In a similar to way to the "Keyboard or joystick" thread, we're to discuss two peripherals for one genre specifically, fighting games.
Being an avid fan to these kinds of game, it's a really important question for someone like me.

First off, it depends on what kind of gaming environment you've been raised, if you're staying home and playing there, you're most likely going to respond control pad. If you're used to play out (mostly in arcades), then your weapon of preference is the joystick. I know that the latter one is a dying breed, with all those online

I'm going to point out the pros and cons of each one of them.
(If you have any more you'd like to point out, just ask me which and I'm going to modify my post, if I feel it's needed)

Control Pad:
+Very cheap to find one of good quality (And comes free most of the time!)
+Everyone is familiar with the control pad
+Lightweight and small
-Hard to perform complex motions
-Can't use all the keys in an easy manner (Mostly for 6 key fighting games)

+Easy to perform complex motions
+Easy to use all the keys
-Quite costy to find one of good quality
-Big and bulky
-Hard to learn

Personally after reading this I'd probably say the control pad, I'm much much more familiar with the control pad but recently I bought a joystick, the Hori Real Arcade Pro 2, it costs quite a big buck. I'm not regretting it though, it's a nice addition to every fighting-game... gamer.
I play a lot with characters, a la Zangief (or if you prefer with 360 degree motions), doing those with a control pad makes your thumb go "Ow!". Even worse is for characters that have 720 motions (i.e. Hugo from Street Fighter 3), it's basically impossible to do that motion with a control pad unless you're jumping or something, the joystick renders that move possible if you're fast enough.
If you have competition at your local arcade, it's better to play with a joystick since every arcade is joystick-oriented.

I'm much more of a joystick person now, I grew up playing Street Fighter 2 with the SNES control pad but I changed my path recently due to the fact that I wanted to go play out and challenge people, it's fun.

One exception comes to my mind, the SSB series, I haven't seen people play with joysticks since it's a rather simple game and can be played with a control pad.