I originally learned to play Street Fighter II on a stock SNES controller. The peak of my fighting game ability probably came during the PS1 era, when I would play Street Fighter Alpha 3 on a DualShock controller. I have a hard time playing fighting games on gamepads now, because my hands start to cramp badly after a few rounds spent gripping a DualShock 3 or Xbox 360 controller. But I still sincerely believe that a good D-pad will always be quicker and less error-prone than even the best joystick lever. The advantage of a joystick is not the stick at all but the buttons, which more naturally utilize multiple digits of the player's right hand, providing faster access and superior ergonomics for when you need to press multiple buttons at the same time or in rapid succession.
As someone who has used both pad and stick, my position generally has been "play with whatever works for you." Contrary to the insistence of some arcade traditionalists (and posers) that joysticks are the only way to go for serious players, a great many pros have enjoyed success at the highest levels while playing on pads. All that said, as someone who principally plays on stick these days, I'm now going to discuss some of the objective shortcomings and frustrations particular to gamepads.
1. There are no uniform standards for gamepads.
It may feel like the opposite is true, since each game console has its own official first-party controller packed in with the system, whereas a joystick is always a third-party specialty item. But, actually, nearly every quality joystick at retail (be it Mad Catz, Hori, Qanba, or Razer) includes the same essential components—specifically, a joystick lever and buttons manufactured by Sanwa Denshi. Meanwhile, the standard controllers for Sony, Microsoft, and Nintendo's consoles all use the respective companies' own patented parts (buttons, D-pads, analog sticks), each with their own distinct qualities and feel.
Moreover, none of today's pack-in controllers are especially well-regarded for use with fighting games. Most fighting games are fairly classical in design—still digital, even as the rest of the industry and technology have shifted focus to analog input—and today's controllers are not built with very much consideration for the genre. Of course, many skilled players have performed astonishingly well with stock DualShock 3 and Xbox 360 controllers. But, recognizing a demand for other options, companies such as Mad Catz and Hori have also released specialty gamepads designed specifically for fighting games. These "fight pads" all feature unique buttons and D-pads, which tend to be made from noticeably cheaper materials than the first-party controllers.
What all this means is that no two gamepads feel or perform the same, and there is no uniform experience when playing on pad. And that's just on consoles; things are even more splintered on PC.
2. Gamepads are difficult (sometimes impossible) to repair or replace.
This is probably the biggest issue with gamepads, and also the reason why the previous point matters.
With a joystick, if your Sanwa lever or buttons break, you can order individual replacements fairly inexpensively (at least compared to buying an entire new joystick) and install them even without a lot of technical know-how. Meanwhile, if the D-pad or a button on your gamepad breaks, none of the various gamepad manufacturers is going to sell you just a D-pad or just a button from one specific controller model. Your only real option is to buy an entire new controller. And if the broken pad was a third-party controller more than a few years old, there's a high probability that you won't be able to find the same model for sale anymore. In that case, you'll have to replace it with a different gamepad, and learn to adapt all over again to the different buttons and feel.
3. Gamepads are not as durable as joysticks.
And this is why the previous point is such a big deal.
This is less of an issue with first-party controllers, but, as mentioned before, third-party products very often feature noticeably cheaper construction. For these third-party fight pads, there are also nowhere near the standards for quality control that Sanwa Denshi has developed over decades for its joystick parts. Unfortunately, even the fight pads from reputable peripherals manufacturers such as Mad Catz and Hori are virtually guaranteed to break with the sort of rigorous use that a serious fighting game player is going to subject them to. The D-pads especially just aren't built to withstand that degree of wear-and-tear. The problem there is that the simple rubber contacts on the inside, underneath the D-pad, get worn away with repeated use. The many negative customer reviews for Mad Catz's last batch of Street Fighter X Tekken FightPads suggest that it doesn't even take that long for it to happen, and these design flaws are maybe why Mad Catz hasn't kept its FightPad in production. They probably know that it wouldn't be cost-effective to design and manufacture a truly premium fight pad, since the high end of the fighting game peripherals market is really the joystick space.
Basically, fight pads aren't built to last, and, once they break, that's when you run into the previous issue of being unable to replace them.
4. Gamepads are difficult (if not impossible) to modify.
Gamepads are difficult to mod for mostly the same reasons that they're hard to repair—there's no way to get the right parts.
Why would you even want to mod your controller, you ask? Well, one popular joystick mod is to make your console-specific joystick compatible across other consoles. This is helpful because, when you enter a tournament, you have to play on whichever platform the tournament organizer decides. Xbox 360 is now the tournament standard for the current Capcom fighting games (because the PS3 versions have acknowledged lag issues), but that is something that has actually changed within the lives of those games (PS3 was standard as recently as Evo 2013). Much cheaper and more convenient than having a different joystick for each platform, one way to prepare yourself for all situations is to have a single joystick that works on both systems.
This is most commonly done by installing a custom PS3 circuit board (available from specialty retailers for $35-60, depending on features) into an Xbox 360 joystick. In that case, you would literally end up with both a PS3 and an Xbox 360 circuit board within the same joystick. That sort of mod is simply not possible with most gamepads, because you won't find two suitable circuit boards compact enough to fit together inside a gamepad.
Instead, if you play on pad, one option is to use an external adapter cable that is supposed to convert your controller to be able to play on the other system. But almost any converter is going to introduce some amount of input lag, which would be unacceptable at the tournament level.
This sucks because pad players also may not even have the option to buy the same style of pad on both consoles. If you play primarily on a stock Xbox 360 controller, that, being a first-party Microsoft product, is obviously not going to be available on PS3.
5. Wireless controllers introduce all sorts of problems when running a tournament.
In contrast to point #3, this is more of an issue with first-party controllers. Both the DualShock 3 and the stock Xbox 360 controller are wireless devices that can interface with the system without any physical connection. This can open the door to such situations as a match being interrupted by an uninvolved party from across the room. You may not mean any harm, but if your controller is, unbeknownst to you, still synced to that system across the room, then any buttons you press may remotely sabotage someone else's game. At a large tournament, there would be no way to realistically keep watch over all the wireless controllers that players in the room might be packing, so quite often such controllers are simply banned outright in order to prevent such situations.
Also, there is the possibility of your wireless controller losing connection or battery mid-match, which would obviously be undesirable. Since the vast majority of joysticks are wired, they do not present these risks.
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The above points only really apply for hardcore players, who are more likely to stress their controllers with repeated use and furious D-pad motions, and who will possibly compete at a tournament level. The vast majority of people who buy and play fighting games will be playing them with stock controllers at only a casual level, and they needn't worry about any such problems. Even back when my coworkers and I were playing Street Fighter III: 3rd Strike on the PS2 during every break at the office, we were doing so just with stock DualShock 2 controllers, which would be handed from one person to the next, as their turn came up. We always played to win, but nobody was so serious about it that they were going to lug their joystick from home in order to become champion of the office. At that level, it doesn't really matter, and, in fact, it would be a pain if every person had to pause to swap in their personal peripheral when their turn came up.
And, even at the tournament level, many pad players have enjoyed great success, including champions like Darryl "Snake Eyez" Lewis (Evo 2010 champion in Super Street Fighter II Turbo HD Remix, current top 3 Zangief player in the world in Super Street Fighter IV: Arcade Edition Ver. 2012) and Alioune Camara (multiple times French and European champion in the Street Fighter IV series, proficient with a wide range of execution-intensive characters). So, again, the most important thing is to find what works best for you.
But if you are a casual pad player considering taking your game to a more serious level, then you should be ready with answers to the potential setbacks I've pointed out. If, for example, you've grown accustomed to playing on a DualShock 3, what are you going to do when you find that wireless controllers have been banned at a tournament you want to enter? Or what if the tournament is held on Xbox 360, and your PS3 controller won't work anyway? (The answer, by the way, for both scenarios is to get a DualShock 1 or 2 instead, along with lagless adapters for PS2-to-PS3 and PS2-to-360.) What if your favorite no-longer-in-production gamepad breaks or is lost or stolen, or it's not compatible with new or different consoles? Will you be willing and able to adapt to playing on a different controller? At least, if you learn to play on a joystick, you'll effectively be "future-proofing" your proficiency at the game, since almost all legit joysticks now use Sanwa parts (or something similar) and perform virtually the same, and it doesn't look like that will be changing any time soon.
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One final important point on the "stick vs. pad" debate is that there are actually yet other options. When I used to play emulated versions of SNK fighting games on PC (because, for a long time, there were no good console ports of King of Fighters and Samurai Shodown), I would use a keyboard (with my right hand for directions, no less), which actually offered quite a number of advantages. When every direction is its own separate key, there is none of the ambiguity and consequent risk of error that accompanies playing on a joystick or even a D-pad. At the same time, a keyboard, like a joystick, lends itself more naturally to playing with multiple digits, allowing you to more easily hit multiple buttons in rapid succession. The concept is carried even further than on a joystick, because, on a keyboard, you can use multiple fingers for not only the attack buttons but also the directions. The disadvantages to playing on a keyboard are 1) a typical cheap keyboard is definitely not built to withstand rugged fighting game play, and 2) keyboards are not compatible with console versions of fighting games.
One other thing to consider with keyboards is that, although being able to use multiple fingers for the directions should make you quicker in many respects, having each direction assigned to its own key also means, on the other hand, that there are no "shortcuts." An old trick on D-pads, for example, is to slide your thumb in a "forward, down, up" motion in order to perform 360-degree techniques such as Zangief's Spinning Piledriver. The Spinning Piledriver actually requires the player to input 6-7 directions (depending on the game) in one continuous motion, so "forward, down, up," which seems to only encompass three directions, shouldn't be enough. The trick works because, on a D-pad, you're likely to incidentally hit the other required directions as you aim for "forward, down, up." And it's useful because it's quicker to think "forward, down, up" than it is to consciously try to hit every direction in between. That doesn't work on a keyboard, however, because the directional keys are separate from one another, so there are no incidental inputs. On a keyboard, every directional input must be deliberate. You've got to mean it! For longer motions, such as 360 and 720-degree maneuvers, that means you're going to have to hit a lot of keys. This is not necessarily a disadvantage, but it takes some getting used to.
A recent contraption is the Hit Box, a controller designed specifically for use with fighting games, which resembles a premium joystick in size and construction, except, instead of a lever, it uses discrete buttons for the four directions. I've never used a Hit Box, but the theory behind the directional buttons makes perfect sense to me. It's the same principle that makes keyboards advantageous for fighting games: having each direction be its own button translates to less ambiguity and greater quickness. Compared to a keyboard, the Hit Box is designed more ergonomically specifically for fighting games (although, with the "up" button located below the "down" button, it is unlike probably any keyboard config you've ever used, so there will be a learning curve), and it features the same high-quality Sanwa buttons as on most high-end joysticks. And it's built to be compatible (and tournament-legal) with PS3 and Xbox 360. I can't personally see myself shelling out that much money for such a specialized peripheral, when I already own multiple joysticks and fight pads while not even competing at a tournament level. But I absolutely approve of the Hit Box concept and would not be surprised to see people start winning tournaments with it.