Magic comes from the Weave. There are a lot of path's to the weave: studying spells (wizard), Gods (cleric), natural forces (druid), pacts with powerful entities (warlock), songs of creation (bard), and even inherited talent from special bloodlines (sorcerer). In the end though, they are all different paths to the same source; the weave where magic comes from and all magic (no matter its source) obeys the rules of "spells" because in the end, they are all coming from the same origin (just through different routes). Think of it like the internet; I can access it from a Windows, Linux,Mac, iOS or Android device using Chrome, Safari, Firefox, Opera, or Internet Explorer, but they all lead to the same internet source.
Psionics is different. It doesn't use the Weave as a source of magic. It pulls from somewhere else.
Y'know what's funny about this? The Weave is a network delivering energy to whomever plugs into it using the right adapter/device/algorithm (components). What you can do with it is consistent, predictable, repeatable - and, with enough training (a wizard level, feat, or even background) anyone can do it.
Psionics, OTOH, are mysterious powers some claim to have that others can't reproduce or explain - or, under the 'psionics are different' theory, detect, or affect.
So spellcasting, in the Realms, is not magic at all: it's just one of those 'sufficiently advanced technologies.'
Psionics, OTOH, actually does have some of the science-defying qualities of magic.
So... psions aren't mortal? If so, they still tap into the same source.
"The Weave" is one of the things I hate about the Realms. I despise that the word "Weave" is even found in the PHB. It's entirely too setting-specific.
I'm not crazy about the high-magic Realms with the Weave and Mystra and her favorite, Elmonster, and, just general uber-NPCs under every rock approach.
But, for whatever reason, they chose it as the default setting, so you have to expect to see it's schticks here and there. Like everything else, though, we're free to toss 'em. As long as Weave-ishness doesn't get hard-coded into how magic works in a what that's too difficult to tease out, though, it's really not an issue.
Arcane magic deals with science-like laws of how to manipulate, well, magic. Depending on the setting, this could be an Enochian "language of creation", manipulation of ambient energy (a.k.a. Weave), or even forcing various entities (read: deities, vestiges, etc.) to do your bidding via loopholes in their service agreement.
Or the material components of your spell and the air you exhale while speaking it being exchanged for the pure energy of the positive material plane to power you spells (1e DMG, if anyone's wondering).
Psionics is somehow internally powered; that "somehow" is the rub. This could be people keyed to the "friction" between different realities (Far Realms), tainted by magical "fallout", capable of directly manipulating the ambient "Weave", or even showing the first signs that humanity has the same divine spark as the gods.
The setting determines how psionics fit. If I were to run a Realms game that included psionics, I can't imagine any other flavor than to say the psion is manipulating the Weave without need of formal spells, regardless of mechanics. Maybe "Mystara's Chosen" is just a cute name for psions.
Sounds reasonable - and flexible.
I still can't see anywhere near as much justification for fluff saying "psionics is different" as for saying "divine channeling is different". You could throw psions on either side of the line, but they have a lot more in common with either arcane or divine magic than the two do with each other.
The biggest factor I see is one that fans of psionics are quick to deny: that psionics is a science-fiction 'bit,' a throwaway detail used to insert magical elements into a genre that otherwise tries to be modern and scientific. Because of that association, the idea that it's "not magic" is actually pretty intuitive, and a little shaky. Psionics in sci-fi is not magic in the sense that it's had traditional trappings of magic removed - the serial numbers filed off - but it is magic in the sense that it does what magic does, both literally, in the sense of the same sorts of effects, and in the literary sense of filling the same function within the story.
So the argument that psionics are 'not magic,' and the argument that they're redundant, and the argument that they have no place in D&D at all, are all really based on the same thing - if one is right, they all are.