The Star Trek franchise employed several styles of Hand-phasers. Star Trek Alpha did not go into details in regards to the amount of shots (clip size) or if the newer models had evolved into more efficient and/or powerful weapons. So, size and shape aside, how did the Federation Hand-phasers compare to each other? I'll accept the best answer with canon info that compares these devices, but would greatly enjoy it if somebody has access to more details regarding additional Federation Hand-Phaser weapons.
There have been a range of Type I and Type II phasers seen in the various Star Trek series.
I'll address the TNG versions first:
The Type I Hand Phaser had a substantially smaller energy capacity than the Type II (roughly 720000MJ versus 4500000MJ) and a smaller pre-fire chamber allowing a smaller range of destructive settings.
Based on the power utilisation figures listed in the "TNG : Technical Manual", the type I phaser could be used on setting 1 (light stun) effectively indefinitely, recharging faster than exhausting. At setting 9 (disruption effects) it would exhaust its power supply in approximately 13 seconds.
The type II phaser would also run indefinitely at the lowest setting and exhaust its power supply at setting 9 in around 1 minute. It was also capable of higher settings up to level 16 (explosive effects) where it would exhaust its power supply in under 1 second but with devastating effect.
By the time of DS9, there had been a slight improvement to the Type II phaser, increasing its power capacity from 4500000MJ to 8790000MJ and a 15% overall improvement in discharge rates. The battery is also "hot-swappable" allowing immediate reloading. Per the "DS9 : Technical Manual"