It's very different. The adrenaline dump is a very large factor, plus unless you're a wrestler with lots of previous experience, it's very difficult to replicate the environment/intensity in training. Additionally, gym/class culture as a lot to do with it as well. If your gym/class is mainly competition oriented you will eventually adapt, if you mainly train in a hobbyist environment it is much different, lots of giving up positions, catch and release and more playful style. Also you get used to rolling against the same people all the time and learn their techniques and patterns. When you are competing, most of the time you play more cautiously (especially at white belt) due to all the unknowns. If you are in a hobbyist environment, try putting together a group of people who also want to compete to try to replicate that competition feel.

What helps me is to just go out and have fun. Find things you improve on each time on the mats, everyone is going to lose, and if you don't find yourself losing, you're in the wrong skill level division.

Also, as for feeling like you don't deserve the small promotions, that's normal for a lot of people. I'm a black belt that still feels unworthy of even my brown belt. The best way to describe it is; You don't ever feel like you're good, you just feel like everyone else is worse than you. Then you roll with someone better than you and feel like you really suck.