simulation is a system that behaves similar to something else, but is implemented in an entirely different way. It provides the basic behavior of a system but may not necessarily abide by all of the rules of the system being simulated. It is there to give you an idea about how something works.
Think of a flight simulator as an example. It looks and feels like you are flying an airplane, but you are completely disconnected from the reality of flying the plane, and you can bend or break those rules as you see fit. E.g.; Fly an Airbus A380 upside down between London and Sydney without breaking it.
emulation is a system that behaves exactly like something else, and abides by all of the rules of the system being emulated. It is effectively a complete replication of another system, right down to being binary compatible with the emulated system’s inputs and outputs, but operating in a different environment to the environment of the original emulated system. The rules are fixed, and cannot be changed or the system fails.
The M.A.M.E. system is built around this very premise. All those old arcade systems that have been long forgotten, that were implemented almost entirely in hardware, or in the firmware of their hardware systems can be emulated right down to the original bugs and crashes that would occur when you reached the highest possible score.