In theory, an emulator is oriented around producing a result (this may mean making acceptable compromises), whereas a simulator is oriented around inspection of state (this usually means being exact).
Adding some anecdata, I feel like emulator is mainly used in the context of gaming, in which case they actually care a great deal about accurate reproduction (see: assembly bugs in N64 emulators that had to be reproduced in order to build TAS). I haven't seen it used much for old architectures; instead I'd call those virtual machines.