The Push press is where the bar is resting on the heals of the hand, with the thumbs back where the fingers are (quite a few heavy benchers prefer this grip also). This allows the bar to be more inline with the forearms, giving better leverage/power and "push" to the movement. There will be a slight knee kick/dip to begin the movement. Can use quite a bit more weight in the Push press than the standard military/front press.
The Military press is where the bar is griped (thumbs aroung the bar) and the bar resting on the palms of the hands, offsetting the direct leverage/power a bit. Usually there is a slight bend in the knees to begin the press. The Military press is not started from behind the neck. That is another exercise called the press behind the neck or behind the neck press (take youe pick of either name to call it). The original idea of the military press was to have both heels together, locked knees and the back ramrod straight. That was considered being done in a military manner (by the military handbood).
In my experience (take it for what that may be worth...if anything) the Push press (like it's brother, the jerk press) is one of the most effected ways to build thick shoulder mass, and in a somewhat shorter period of time. Can also effect the traps/upper pecs/triceps.. Very heavy weight can be "pushed' overhead, sooner than you think. You should be buying larger size shirts if approaching, with serious intent, the Push press.
Sitting , while overhead pressing, is more suitable to the military/front press it's self. Because the body English from the knees/hips is taken away. If you are using one of what may be called shoulder/pressing benches, you are really doing a high angle incline press. If you are doing overhead presses with a 90 degree angle sit down bench, then you are doing a real overhead press, influencing the shoulders more directly.
DB presses are also a superior movement for the whole shoulder girdle. Suggest them being done standing and one arm at a time, for a better focus on this exercise. Bracing the free arm, while overhead pressing, seems to help with this exercise greatly.
Training for partial reps (prefered on a PR), in any form of overhead pressing, can build exceptional shoulder and give unbelieviable results in size and power to some. Good Luck..