I don't think emulator and simulator can be compared. Both mimic something, but are not part of the same scope of reasonning, they are not used in the same context.
In short: an emulator is designed to copy some features of the orginial and can even replace it in the real environment. A simulator is not desgined to copy the features of the original, but only to appear similar to the original to human beings. Without the features of the orginal, the simulator cannot replace it in the real environment.
An emulator is a device that mimics something close enough so that it can be substituted to the real thing. E.g you want a circuit to work like a ROM (read only memory) circuit, but also wants to adjust the content until it is what you want. You'll use a ROM emulator, a black box (likely to be CPU-based) with a physical and electrical interfaces compatible with the ROM you want to emulate. The emulator will be plugged into the device in place of the real ROM. The motherboard will not see any difference when working, but you will be able to change the emulated-ROM content easily. Said otherwise the emulator will act exactly as the actual thing in its motherboard context (maybe a little bit slower due to actual internal model) but there will be additional functions (like re-writing) visible only to the designer, out of the motherboard context. So emulator definition would be: something that mimic the original, has all of its functional features, can actually replace it to some extend in the real world, and may have additional features not visible in the normal context.
A simulator is used in another thinking context, e.g a plane simulator, a car simulator, etc. The simulation will take care only of some aspect of the actual thing, usually those related to how a human being will perceive and control it. The simulator will not perform the functions of the real stuff, and cannot be sustituted to it. The plane simulator will not fly or carry someone, it's not its purpose at all. The simulator is not intended to work, but to appear to the pilot somehow like the actual thing for purposes other than its normal ones, e.g. to allow ground training (including in unusual situations like all-engine failure). So simulator definition would be: something that can appear to human, to some extend, like the original, but cannot replace it for actual use. In addition the pilot will know that the simulator is a simulator.
I don't think we'll see any ROM simulator, because ROM are not interacting with human beings, nor we'll see any plane emulator, because planes cannot have a replacement performing the same functions in the real world.
In my view the model inside an emulator or a simulator can be anything, and has not to be similar to the model of the original. A ROM emulator model will likely be software instead of hardware, MS Flight Simulator cannot be more software than it is.
This comparison of both terms will contradict the currently selected answer (from Toybuilder) which puts the difference on the internal model, while my suggestion is that the difference is whether the fake can or cannot be used to perform the actual function in the actual world (to some accepted extend, indeed).
Note that the plane simulator will have also to simulate the earth, the sun, the wind, etc, which are not part of the plane, so a plane simulator will have to mimic some aspects of the plane, as well as the environment of the plane because it is not used in this actual environment, but in a training room.
This is a big difference with the emulator which emulates only the orginal, and its purpose is to be used in the environment of the original with no need to emulate it. Back to the plane context... what could be a plane emulator? Maybe a train that will connect two airports -- actually two plane steps -- carrying passengers, with stewardesses onboard, with car interior looking like an actual plane cabin, and with captain saying "ladies and gentlemen our altitude is currenlty 10 kms and the temperature at our destination is 24°C". Its benefit is difficult to see, hum...
As a conclusion, the emulator is a real thing intended to work, the simulator is a fake intended to trick the user.