Generally speaking, the Mach Bike has far more applications than the Acro Bike. Despite the tricks that the Acro Bike can do, it only accesses a few items in areas that otherwise aren't important enough to make it any more useful. The only items that can be accessed with an Acro Bike before obtaining Waterfall in Emerald would be a few Sitrus and Leppa Berries, as well as a Calcium, on Route 119.
There is also Area 4 in the Safari Zone, requiring the Acro Bike to reach, which contains its own Calcium, and also being the only areas where Phanpy and Heracross are encounterable. As it's not hard to backtrack to either area after obtaining Fly, I don't think you'd be hard-pressed on missing out on the items or the Pokemon blocked off from accessing without the bike.
However, it's worth noting that, when bunny hopping with the Acro Bike in tall grass, it's possible to encounter wild Pokémon without it counting towards steps for the Safari Zone or Repels. This allows for encounters in the Safari Zone or Repel tricking with far fewer interruptions, as well as avoiding the additional cost. Bunny hopping also makes for getting into random encounters with multiple systems being played at the same time far easier, whether it's for shiny hunting or anything else, as it simply requires you to hold the B button.
Overall, I'd recommend sticking with the Mach Bike for most of the game, and swapping to the Acro Bike temporarily if you plan on doing frequent random encounters or you're interested in getting the items/Pokemon you missed out on without it.
