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The eternal debate. Which is better? The short answer, whatever works for you. The long answer? Well it's a little bit longer.
Let me start by explaining my experiences. When I was a kid, I went to the arcade. My motor skills were so poor that I never really noticed any difference between pad and stick. So for years I played Street Fighter on the Genesis and then PlayStation controller (I even went as far as to buy a PS controller converter for the Dreamcast). I didn't know any different and for a time things were good. I could beat my brothers and friends who came over, and arcades were dying out so I never really had to worry about playing on stick. The only time I had an issue was playing Marvel vs Capcom at a local community college; I was getting whomped on by everyone. It hurt too because I knew I could play better. However; I only went to that school twice for some event in High School and never had to think about it again.
Flash forward a few years: Street Fighter IV came out. There happened to be an arcade, in a bowling alley, a couple cities over where I would go to play. I was so exited at the time that I didn't even mind losing most of my matches because I couldn't use a stick. When I got tired of losing, knowing I could win, I bought an Hori EX2 and got to work practicing. It took about a month, but I could do almost everything I could on Pad. I was pretty psyched, so I went down to the arcade and won a few matches (the high point was beating people even though my side of the machine's stick was broken). I felt great and got excited when I heard about the Street Fighter National Tournament that was being hosted by Gamestop.
I redoubled my efforts at mastering the stick and started to feel really confident (thanks to playing a ton of HD Remix). I rolled up to Gamestop and was ready to win. One catch, it was entirely on Xbox 360 pads, not even the fancy Madcatz pads, just regular 360 pads. I didn't really think about training on those, so I lost in the second round (also 360 pads suck for Street Fighter). All that work for nothing.
This brings me back to Pad vs Stick: Most tournaments are run off console, so just go with what you like. I know from personal experience you can work faster on a stick, but it doesn't seem like a huge difference. Also, if you show up at a tournament some people might try and clown on you, but that will stop once you win. Even showing up with a pad can work in your advantage because your opponent will expect you to suck, until you FDAC into Ultra on their face. If you're playing over Live or PSN, no one knows what your playing with, so just settle in with your weapon of choice and get to work. Right now, I like stick more than anything else. However, it was a lot of work to get to this point and I'm not a tournament level player, so in the long run it won't even matter. All that really matters is that you enjoy playing and can throw a beatdown when needed.