Oooh, good question!
You're talking about quantity, so we have to identify "how many" each expression represents.
Let's see, and please comment on my comments if you have any comments.
A bunch generally means a lot. Not a whole lot, but not a few either. If the game is touch football, for example, to say there was a bunch of kids playing might mean between 10 and 20 kids.
If there was a bunch of players dominating a game, you might think of the same number (10 - 20) taken out of a larger number.
A handful is about the number of objects or items you can fit into a closed fist. What do people usually have in their hands and hold onto with a closed fist? Maybe coins, some paper clips, a group of pencils, some popcorn. With that in mind, you might be thinking of a number anywhere between 3 and 7. Any more than that and you're getting into "a bunch" territory!
In the example given, a game being dominated by a handful of players might mean just 4 guys or gals out of a relatively higher number.
Hope you enjoy!