Why is it whenever I come to use Audio Snap it won't do a simple task I give it?
I've recorded a rhythm guitar part when the project was 102BPM.  I changed it to 110BPM.  I want to change the guitar part.
I highlighted the clip, and followed the Help instructions, which are:
 
To sync an audio clip to the project tempo
1.
Select the audio clip(s) that you want to follow the project tempo.

2.
Press A to open the AudioSnap palette.

3.
Click the Clip Follows Project Tempo button on the AudioSnap palette.

The clip is synchronized to the project’s tempo map
 
No it isn't.  Nothing changes.
What am I doing wrong?

You need to enable Clip Follows Project in the default Autostretch mode (option in the dropdown) before you change the tempo. In this mode, nothing happens until you change the tempo at which point the clip stretches/compresses to follow the change.
 
There are also options to sync a clip to the project after the fact, but it generally requires massaging the clip's tempo map and changing the Clip Follows Project mode (options in the dropdown to the right).