WAV:
stores the raw audio waveform. Pretty much nothing special you need to do with it to get it to make noise, just send it to the audio card and it'll play
MIDI:
Stores (essentially) the "sheet music" for a song. When to play what note, for how long, at what volume, and which instrument. A wav file needs 44100 samples per second to make audio (at 4 bytes per sample, that's 176kb per second). A midi file is much more compressed because it's not storing sound, but instructions on how to recreate that sound.
Of course, most programs have their own "midi sound generators". A midi file can be read in by 2 different programs and recreated with completely different instruments in each one (although most try to find the closest match).
As a result, you should really have the program making the midi file export a wav or mp3, otherwise it will sound different