Not moving legs with the lift for the overhead part is either a push press (dip slightly at the knees, start press as you extend to full leg lockout, which gives a boost on the press part and allows more weight to be handled), or, a power jerk, which is where you dip slightly at the knees and then rise up to stand erect for the overhead press part, but as the bar starts to move upward quickly on rising, you dip back under it again ala the "jerk" part of the movement so that there's less actual shoulder/triceps pressing done to get under the bar. Not a great explanation, but should show a bit of difference between the two. Push pressing is usually a slower movement much like any regular overhead press, where the jerk is intended to be done split-second, extremely fast and athletic, not a "brute strength" type of lift but one that's far more technical.
So, a clean (power clean, to be precise) and push press looks something like this:http://youtu.be/doz6O1MKMcQ
...And a clean and power jerk looks like this:http://youtu.be/XD1yTElnC1c
The speed and technique of the overhead lifting part being the difference, as in competition, a jerk will be disqualified if the bar does not go to full lockout at arm's length overhead in one fluid motion with no slowdown. Now that the boring technical part is out of the way....
The power clean part will work everything that it does when deadlifting (hamstrings, glutes, lower back, middle back, upper back), the overhead part will be for shoulders, triceps (primarily with push pressing or strict overhead pressing with no dip before the press) and core stability via the abs and obliques.
That about sums it up!