Here's the difference in a nutshell (as seen by rharv)

When you use ASIO, the idea is to play a keyboard and have the MIDI from a keyboard come into the computer and generate a softsynth. Or to record audio and use the realtime effects on the input. Example: sing into a mic and hear the reverb from a DXi or VST effect in real time.

If you don't use those features you don't need ASIO. You may want to, and on some systems these drivers actually work better, but this helps determine the need.

The reason: when the MIDI comes in (or the audio) the computer is tasked with creating a sound right away. The sooner the better! So ASIO runs in the system with much lower latency (delay) because it makes itself a priority. By doing this it adds stress to a system. Some systems and drivers are fine with this, some not so much.

MME, on the other hand, doesn't respond as quickly. This is because the audio buffer it uses is much larger, so it doesn't tax the system so much. It still creates the same sound, but just does it with less stress on the machine and a bigger buffer.

This buffer comes into play in many ways, but most are not needed by many users except in the above two examples.
Some others would include:
Response to a mixer interface (including the program mixer)
Milliseconds of time between hitting play and the file starting (buffer load)

Things that require immediate response.
Otherwise, they are indistinguishable in the end.

I do not work here, but the benefits are still awesome
Make your sound your own!