Well, you can catch mice for your sand cat. I know it's usually the other way around, but I guess you have to prove to be better than him at hunting to earn his respect.Is there even any benefit of capturing the small animals? The bug catching seems pretty underutilized atm. I never even bothered crafting the net. Fishing isn't much better, so far really only seemed like gold fish are required for anything.
A few other monsters like to eat live insects, but there is usually a more accessible alternative, like bait.
And if you catch a live crab, it can be used as an ingredient for a quite good cooking recipe.
But overall, catching things with a bug net seems to be something collectors would like, and not much of a gameplay feature.