The VIP system (I covered more or less everything, except something about proofs of infinity, no idea what it is):
-You can use diamonds to buy gold, stamina and arena tickets more often
-You have a bigger shop (but price are way too expensive anyway), but you don't get better deals.
-Bigger friendlist
-Can use more often the world channel
-You can reset the hard mode (and you have more uses/day)
-You get commodities each day (instant stage tickets, for instance)
-You can make instant several things in the game.
Everything is more or less unlocked at VIP 4 (you get the multiple instant clear, which is pretty useless), after that, you only gain bigger numbers on the amount of time you can buy something.
The VIP 3 is very nice however, and it looks like it's not that hard to achieve even in F2P (with the free vip/day, you can achieve it in one month, and it's not counting the free nutaku gold). If you intend to spend a little bit, you may want to aim for that. It can change after the nutaku release, however.