For starters, there's the reasons for a hero to maintain a secret identity ever since the Baroness Emmuska Orczy created the Scarlet Pimpernel, it causes no end of frustration for the villain[s] when they don't know who the hero is or where he may be at any given time. And of course the secret identity also shields the hero's family & friends from reprisals; while He-Man is the most powerful man in the universe but he can't be everywhere at once.
As for Prince Adam not staying He-Man all the time, it's hard to say as we don't have much to go on. Perhaps it's a matter of learning to fully use the Power of Grayskull or perhaps Adam simply chooses not to remain He-Man all the time as the relationships with his friends & family will irrevocably change once they know Prince Adam is He-Man.
As for Adam & He-Man looking largely the same, I think budget decided that rather than creative choice. If I recall correctly, Filmation ended up using the same base character model for Prince Adam and He-Man to save on costs but they originally wanted to use a different weaker appearing depiction for Adam. In that regard the MYP series is reflective of what Filmation had wanted to do nearly 20 years earlier.