Shell depth
The longer a shell becomes, the more resonant it becomes.
It also becomes less sensitive and it's full response takes longer to accomplish.
Presence is increased as well.
However, added length can actually colour the drum darker and tends to create a mellower tone.
The Jazz guys tend to stick with the shorter lengths because of the sensitivity and presence factor.
Power Depth
Once upon a time, (for the most part) drumset shells were available in only one depth.
For instance, a 12" tom was ALWAYS 8" deep.
Then, starting in the 1970's, manufacturers started experimenting with longer lengths, to see what their impact on the market would be.
Then along came the 1980's, and with it the Heavy Metal movement.
The aggressive look of the longer shells appealed to this crowd and the "race" was on!
Basically, the "accepted" depth was 2" more than the "traditional" length (so the 12" tom becomes 10" deep) and these lengths were widely marketed as "Deep" Shells, sizes, depths, etc.
The "Metal-heads" yearned for an even more aggressive look, thus shells became even deeper.
These deeper shells were often marketed as "POWER" lengths, sizes, depths, etc.
Nowadays, it seems that anything deeper than a "traditional" or "FAST" depth (1" over traditional) is known as a "Power" depth and the "Deep" nomenclature is becoming lost to history.