Folk style wrestling is a sort of umbrella term that refers to the local style of wrestling unique to any given culture. But in the context of this discussion and most MMA discussion, folkstyle is collegiate wrestling in the United States.
I once posted this in another thread its pretty comprehensive.
Folkstyle does not reward pushing your opponent out of bounds. Freestyle does.
Folkstyle encourages way more prolonged engagements between wresters. If you can ride/keep an opponent wrapped up and continuously return him to the mat for a minute, you get an extra point. You cant lock your hands together during this control. On the flip side, if you escape this entanglement you also get a point. In freestyle the exchanges are much different and much quicker. There is no reward for, and no opportunity to pursue, prolonged bouts of controlling your opponent.
In folkstyle a TD is a TD, regardless of how dynamic it is. It is 2 points. In freestyle a "regular" TD like a double where your opponent is able to immediately go belly down is worth 2 points. Take your opponent off his feet and directly to his back, that's worth 4 points. In freestyle if you hit a big TD or throw and get your opponents hips above your own and that's worth 4. Get his hips up to your shoulders and that's worth 5.
A pin ends the match in both styles. There are near-falls in folkstyle which are rewarded and there are back exposure points in freestyle that are rewarded.
Lose a challenge in freestyle and you surrender a point. Not like that in folkstyle.
Folkstyle matches are 3 minutes 1st period followed by 2 two minute periods. Freestyle matches are 2 three minute periods.
In folkstyle, a 15 point advantage ends the match. In freestyle it is 10 points.
Greco roman has pretty much the same rules as freestyle but you cannot attack the legs. Because of this, Greco guys tend to be really good at throws and what is called par terre, which is the mat aspect of Olympic style wrestling.