@Kettlebelephant pretty much hit the nail on the head but I'll my .02 with my experiences with Jerks.
In the Military Press there is NO leg drive just straight upper body strength. The Push Press incorporates leg drive at the start to get the weight off of your shoulders and use your triceps to lock the weight out overhead. The Jerk, as mentioned above, is ALL legs. With a Jerk you drive the bell/bar from the rack and then dip under to catch the weight. The Jerks is, for me, very easy to troubleshoot in the fact there shouldn't be any "Grind" to the lift except maybe fixation at the top. If you "pressed" the weight during the jerk and had to grind the lockout you did it wrong and didn't dip under properly. Whenever I do KB C&Js my quads will get a pretty good pump as the volume and weight increases.
If it tells you much more weight you can KB Clean & Jerk than KB Clean & Press, I will share my experience using The Beast (48 kg / 106 lbs) over the past couple of months:
KB Clean & Strict Press: 1 very tough rep with each arm. A Max Effort!
KB Clean & Jerk: 5 Ladders of 1,2,3 reps with minimal rest and could add a 6th Ladder or add 4 reps to the Ladder.
As you can see, The Jerk allows you to move a lot more weight for more reps than the Strict Press. I'm sure the Push Press is somewhere in the middle but for my goals and purposes I don't do them. I'm a bigger fan of Barbell Military Presses but love the Kettlebell for Jerks!