This answer was correct as of the question's asking. As later revisions to the UA Mystic came along, they invalidated this answer. This answer contains the latest official description of psionics.
In the D&D podcast of July 10 Mike Mearls describes it, starting at about 14:00, as
The power of the mind. It is a power source that comes from within... it derives from the opening of the mind's eye....
He then goes on to discuss how the Far Realm in D&D cosmology "bumps into" our planes and "can twist reality a little bit." Continuing,
What happens is, if you're psionic your mind's eye is open and you see the underlaying structure of this reality in a way that other people don't. And by drawing on your willpower and your own mental energy you can exert changes to the structure of reality around you- or you can perceive things.... Or you can understand your body and manipulate your muscles beyond what would normally be possible.
He goes on to stress the separate nature of psionics and magic, describing how they've intentionally renamed many psionic abilities to avoid spell-names.
As for the distinction with ki, the class description of a monk specifies that monks have made a specialized study of "an element of the magic that suffuses the multiverse--specifically, the element that flows through living bodies." (PHB p.76)
So monks are using one form of the magical energy that's native to the multiverse; mystics are in tune with things coming from farther out.
Related reading (from comments):
Psionics (role-playing games) - Wikipedia
Why do psionics suck? - AngryGM
As written, the bow creates its own ammunition "[w]hen you draw back its string and fire." As such, you cannot fire any other arrows you might happen to have on your person: the one the Moon Bow creates would get in the way.
That said, it would not be game breaking to allow the Moon Bow to fire magical arrows. Yes you will do extra damage but you are expending a magic item and a spell slot to do so.
They should stack.
Assuming you are using version 3 of the Mystic Class the +3 ability is Psionic Weapon:Augmented Weapon, and the +4 is Hone the Blade class feature of Order of the Soul Knife, I see nothing immediately that would preclude you from stacking these bonuses. It would cost you 12 points, and the former is a 7 pt ability, meaning you would not be able to use it until 9th level.
Your DM may be thinking of the rule about figuring things like Armor Class, where you can only have one formula, and you can choose the more favorable one. In general, in 5th Edition, Formulas may be used once, but bonuses do indeed, stack.