What is the difference in compression and rarefaction between a loud sound and a soft sound?

The difference in compression and rarefaction between a loud and a soft sound is the change in density of the medium conducting the sound. We know that sound is a mechanical wave, and it requires a medium through which to travel. (Sound transfers its energy into the medium to propagate.) Let's do an experiment taking air for the medium and look closely at what is happening.

We've got an amp connected to a speaker and a constant signal being amplified. The signal has a characteristic frequency and amplitude. Air is being compressed and rarefied as the speaker cone moves out and in (respectively) to create the sound. Now we'll turn up the volume and look again.

What we saw before will change in that the speaker cone will move farther out and in than it did before. It will still move at the same rate as it did because the frequency of our signal did not change. But because it is moving further out, it will compress air "more" than it did before the volume was increased. There will also be a correspondingly greater "decompression" of air when the speaker cone moves back. Air density will be greater in the compression phase at the new amplitude than it was. And air density will be less in the rarefaction phase at the new amplitude than it was.

The difference in compression and rarefaction between loud and soft sound is the relative density of the medium during these portions of the wave. Certainly if the sound is moving through a liquid or solid, there will be a much lower change of density of the medium as liquids and solids are largely incompressible by sound. But the idea is the same as it is in the model of sound moving through air.