Mitama pretty much determine your "spec", if you want to call it that. Whether you're a tank, heals, utility/support, pure damage, etc. The main difference would be the four class abilities you have depending on your spec/Mitama. On the bottom right of the screen, you have four abilities, each corresponding to square, circle, etc. Those are what change based on what mitama you have equipped.
Also, DEF Mitama, the ones that make you a tank (taunts and such) also give you a shield bar over-and-above your health bar. No idea if any others do that, since I've only played DEF and HLG so far. Also, as far as I know based on my time with the demo, your weapon combos and general playstyle of each weapon do not vary based on the Mitama you have.
There are weapons that are better suited for certain Mitama (Fists + DEF, Rifle + SPT, blah blah blah), but you can generally use whatever weapon you want with whatever Mitama you have equipped. For example, I use the fists weapons with a healing (HLG) Mitama and I try to play like a front-line fighter/healer type, sort of like a tanky monk. Other times I just go DEF and play punchy-punchy blocky-blocky.
The image in the OP gives great descriptions of each one if you haven't seen it already.