Both are models of an object that you have some means of controlling inputs and observing outputs. With an emulator, you want the output to be exactly what the object you are emulating would produce. With a simulator, you want certain properties of your output to be similar to what the object would produce.
Let me give an example -- suppose you want to do some system testing to see how adding a new sensor (like a thermometer) to a system would affect the system. You know that the thermometer sends a message 8 time a second containing its measurement.
Simulation -- if you do not have the thermometer yet, but you want to test that this message rate will not overload you system, you can simulate the sensor by attaching a unit that sends a random number 8 times a second. You can run any test that does not rely on the actual value the sensor sends.
Emulation -- suppose you have a very expensive thermometer that measures to 0.001 C, and you want to see if you can get by with a cheaper thermometer that only measures to the nearest 0.5 C. You can emulate the cheaper thermometer using an expensive thermometer by rounding the reading to the nearest 0.5 C and running tests that rely on the temperature values.