The Mute button closes the first VCA of the channel, thus preventing the sound from being routed through all the other VCAs (FX sends, pan, main fader).
The Solo in Place button isolates some channels and mutes the others. However, the mute function it performs is different since it affects the channel after the fader: the sound will still be routed to any pre-fader VCA such as Direct Output and FX Sends (when set to Pre-Fader Mode).
In other words, the Mute button works pre-fader, while the Solo in Place works post-fader.