Just IME, but...
Some waxes do seem to leave "residual artifacts" once they're dead. Souveran and Malm's liquid come to mind. This is seldom a big deal for quite a while; you can just apply new wax over the remnants of the old with no problems but sooner or later things need cleaned off. Other waxes (e.g., #16, Collinite) don't seem bad at all in this regard and you can just add more for a long, long time as needed.
I *would* suggest maybe a light (*very* gentle) claying with something like Sonus Green to get the worst of any [stuff] off before refreshing the wax that way, but I'll admit I don't always do it.
I don't always clean things off before applying a different wax either, but that's just me.
Sooner or later, you really should do *some* kind of real paint cleaning (AIO, paint cleaner, mild polish...something), but it's often possible to put that off for a surprisingly long time. Just depends what the vehicle has been through and how clean it's kept.