There are at least two differences:
1) Standing presses never have the feature of having the back supported while the upper body reclines somewhat. That feature can greatly boost the weight that can be lifted while not necessarily meaning any greater if even as much of what I would call real strength. It can disguise functional weakness that prevents a good lift without that support. Not that such weakness must exist, but it can exist and not be revealed if this is the only style of pressing used.
Of course seated presses can be done with the upper body vertical and no back support, in which case this difference is eliminated. However it seems this is not so often done as is the slightly reclined, back-supported style.
2) Standing presses can be done either with leg movement assisting the lift, or not, as you note. Seated presses must be done without leg assistance. The inability to employ leg drive can be either an advantage or a disadvantage according to the situation.
I don't believe balance is a limitation to a person with reasonable skill in standing presses.