I will say there are two main downsides to using bombs for trees.
1. you don't get any forage xp, if you're working towards forage level 10 something like 100 trees can make a huge difference
2. the bigger issue is that you don't get tree seeds from trees chopped down with bombs, basically making it impossible to create a renewable source of wood. If you bomb most of the trees on your farm you can be left with few maple and oak tree seeds which are invaluable to tap later along in the game for recipes such as kegs and beehives.
I would recommend bombing to lower the tree hp then going in and finishing them off with the axe, this ensures you get the best of both worlds (plus it's super satisfying for each tree to fall in a single axe hit)