The way the two store audio is night and day different. Simply put, a WAV file (which is PCM audio) is storing much more data than the MP3 file is.
PCM audio quantifies the pressure at each sample. There are many samples taken. CD-quality audio uses 44,100 samples per second, per channel. The red lines here are samples:
Graph from: http://en.wikipedia.org/wiki/Sampling_rate
MPEG-1 Layer 3 (and many other lossy audio compression codecs) use a different method for encoding audio. Rather than measuring pressure over time, they measure frequency components over a time. MP3 encoders determine which frequencies are present in a signal for a short period of time, called a frame. A frame may be 576 samples or so long. For those samples, they playback a set of frequencies.
Now, all of this is overly simplified. MP3 has a lot of nice tricks to filter out frequencies that will be masked by others, and some nice smoothing of the playback to make it sound close to the original.
You might find my answer here a bit more in depth: https://video.stackexchange.com/a/635/129