I think it is more about design, emulation mimics what something does. A simulator replicates what something does.

It is a tiny distinction, but generally I'd say that a simulator tries to accurately replicate what happens on an electrical level as good one can do.

While an emulator just does things as a black box ... input produces the expected output using whatever.

You could compare it to that an accurate simulator of a 74181 tries to do it by using AND/OR/NOT/... logic, but an emulator does it using "normal code".

In HDL you have a similar situation between structural, behavioral design ... structural is generally based on much more lower level logic (eg., AND/NOR/.. gates ...), and behavioral on higher logic (addition, subtraction ...).

"100%" accuracy can be achieved with both methods.