Jon Strickland of the Billy Wicks Foundation and American Hook Style Wrestling, the American lineage of Catch Wrestling explains the differences between Folkstyle and Catch Wrestling in his own words:
Like night and day really. Besides the obvious hooking, Catch is almost all mat wrestling while today Folkstyle focuses mostly on the feet. Stalling is illegal in Folk but its the core of the Catch game. In Catch Wrestling you aren't driven by points and that makes a huge difference.
Now throw in the time limits and periods of Folkstyle. Once you do that, you have to think about gripping. Folk is more, what grips can you use. In catch there no illegal grips. It reminds me of Judo and Jiu Jitsu. Now people argue they're the same arts, but in the US that's not true usually. Judo is standing jacket wrestling where Jiu Jitsu is mat grappling.
In catch you could take your opponent down 5000 times in 45 minutes and not really have accomplished much, except hopefully injuring him from such, or mentally breaking him. He could hit you with a hook, (submission), on the 5000th take down and win the match. Its common to see folks claiming to know Catch Wrestling, when in reality they're developing a new definition to full fill their own self interest.
If you have been doing catch for a few years then you should be able to wrestle. If you were encouraged to seek out an amateur coach for wrestling, then you didn't learn Catch but maybe a small portion of it. If someone comes to me to learn that has no experience, I spend the first few months teaching wrestling only, and that's not Folkstyle but the complete style of Catch Wrestling. Later we go to the hooks and other things like rips and grinds.
Guess I and few others were lucky. We didn't need a movement to join or fly around the world. Though I did some traveling in search of the style early on. No, we had a full-time gym with daily access and great sparring partners. Yes the first few years I got stomped on a nightly bases. If you need to win I'd recommend another pastime. Highly recommend to anyone doing it because they enjoy it. I too wanted to just win in the gym, but I didn't improve that way. Have fun because you learn more, and better when your mind is calm.
Train in authentic Catch As Catch Can Wrestling at: www.billywicksfoundation.org