Semantics aside, what are the distinguishing features of psionics that set it apart from "spellcasting magic?" As others have pointed out, there is not a lot of room to have it do things that spellcasting magic can't do, because spellcasting magic can do just about anything. But psionics could do certain things better, or in different ways.
Well, one point: is it actually different from spellcasting mechanically? Does actually using a psionic power entail a moment of concentration and any of the following: a word, an action, or some catalyst material?
There's examples that do. 3e psionics. 4e psioincs. Do we change it so it doesn't and thus doesn't use the spellcasting mechanic? Or do we use the spellcasting mechanic and say that a psionic class does this
differently somehow?
I would say the distinguishing traits of psionics should be:
No external tools, ingredients, or actions. Using psionics is a purely mental activity. You can do it--at full power--while chained, gagged, and naked. (This means none of the New Age crystals-and-chakra stuff.)
Okay, so it doesn't use components. Can we use spellcasting mechanics and say that psions get some feature/ability to ignore components? There's certainly precedence for this in 5e already (check out the gith races). Or do we need/want a whole new mechanic?
Psionics does not create free-standing effects. It does not summon creatures or create objects, and any ongoing psionic effect requires concentration. There is no such thing as a "psionic item."
....sounds like you're not a fan of 3e psionics.

Fair enough. But this is mostly an aesthetic point, not a mechanical treatment - it doesn't speak to what the character actually
does to make the psionics happen.
[*]Psionics is not dispellable. Not only do dispel magic, counterspell, antimagic field, and the like not work, but there are no psionic equivalents of those things. If you want to stop a psion from doing something, you have to target the effect, not the psionics; for example, physically holding still an object that the psion is trying to move with telekinesis.
More points for not using spellcasting, but there are magical effects that don't rely on spellcasting in 5e already (the aforementioned monk's agelessness). Is it always on or does it require some action to do? If it requires some action, can the action be interrupted? Can concentration be broken? Or is psionics "unstoppable," without counter-measures? This could be a pretty rich vein to mine in how it's different from spellcasting, and though we'd need to be careful with the effects we could grant, it could be pretty deep.
Psionics ignores physical barriers. Where psionics is concerned, there is no difference between "line of sight" and "line of effect." A wall of force blocks spells, but psionics can go right through it. Furthermore, most psionic powers can be used "over the wire" of a scrying effect, either magical or psionic.
Potentially mechanically problematic, but possible with tight control of the effects psionics is capable of. Does that mean that I can't charm someone I know is behind a door?
Psionics can be developed by non-specialists. Someone with the Wild Talent feat (I assume this will be a feat) has access to the same array of powers as a full psion. The psion can just put a lot more oomph behind those powers, and can master a greater number of them.
A skill-like system would also satisfy that - anyone can use Insight, but a psion can augment their Insight check to do things like read minds or whatever.
Psionics can't compete with magic in raw power. Psionics ignores a lot of the restrictions and countermeasures that magic-wielders have to deal with; the flip side is that magic-wielders pack a harder punch. Put a psion and a wizard of the same level in an arena, and the wizard will usually trounce the psion.
Another aesthetic point, not really relevant to if it's spellcasting or some other mechanic.
But it seems on the whole like you'd want psionics to NOT be spellcasting. Okay.
What does it look like in the world when a psion uses an ability? Do they activate it with an action or is it always on? Is it something that must be concentrated on? Maybe it varies with the kind of ability used?
SkidAce said:
In the example he gave, and in the mind pf those who like the "science sounding names", it is not magic. Its an advanced ability unlocked by brain or body power. Maybe a magic person can fly, and a psionic person can fly...but that doesn't mean they are both magic (or both psionic for that matter).
If a human being flies due purely to their own will, it's pretty clearly magical. It might be planar energies or breaking the laws of physics or a pact with a demon or the granted ability of a god or just thinking happy thoughts - that's still magical, it's still magic, it's clearly not natural.
It could also be nanobots or anti-grav boots or jetpacks, and it would still be pretty magical, though I'd probably want different mechanics to embody that (like the current magic item mechanics!).
It sounds like when people say "it's not magic" what they mostly mean is "it's not like what a wizard or a cleric does" but neither is a bear totem barbarian's ability to carry twice their normal load, but that's still magical, it just doesn't involve casting a spell. "Magic" in D&D isn't just what wizards or clerics do.