How is CPM Modulation Different from PSK?
My understanding of the difference is as follows:

With CPM the change in phase happens over the entire time of the symbol. With QPSK it happens at the beginning. Agreed, it's not an abrupt change from one phase to another, but it still takes considerably less time than the time for the symbol. 

So in QPSK there is a change phase, followed by a hold phase. In CPM there is just the change phase, and no hold phase.

Say a symbol lasts for 4 wavelengths, and you want to change from 0° to 90°. In QPSK the first wave would be the transition from 0° to 90° (it would typically take less than 1 wave to change phase), and the next 3 would be holding at 90°.  With CPM each wave would be a different phase from 0° to 90° - say 22.5°, 45°, 67.5° and 90°.

Consequently, CPM encodes extra information that's not present in QPSK.  If you go from 0° to 180°, which way did it go? Clockwise, or counter-clockwise? QPSK has no way of knowing that.  With CPM that direction of change is also encoded, along with the amount of change.