Emulation is the process of mimicking the outwardly observable behavior to match an existing target. The internal state of the emulation mechanism does not have to accurately reflect the internal state of the target which it is emulating.
Simulation, on the other hand, involves modeling the underlying state of the target. The end result of a good simulation is that the simulation model will emulate the target which it is simulating.
Ideally, you should be able to look into the simulation and observe properties that you would also see if you looked into the original target. In practice, there may some shortcuts to the simulation for performance reasons — that is, some internal aspects of the simulation may actually be an emulation.
MAME is an arcade game emulator; Hyperterm is a (not very good) terminal emulator. There’s no need to model the arcade machine or a terminal in detail to get the desired emulated behavior.
Flight Simulator is a simulator; SPICE is an electronics simulator. They model as much as possible every detail of the target to represent what the target does in reality.
EDIT: Other responses have pointed out that the goal of an emulation is to able to substitute for the object it is emulating. That’s an important point. A simulation’s focus is more on the modeling of the internal state of the target — and the simulation does not necessarily lead to emulation. In particular, a simulation may run far slower than real-time. SPICE, for example, cannot substitute for an actual electronics circuit (even if assuming there was some kind of magical device that perfectly interfaces electrical circuits to a SPICE simulation.)
A simulation does not always lead to emulation —