Psionics should not feel like magic with different fluff.
For me, the lack of psionics in 5E is very disappointing. I run a campaign setting that I built in the 1980s. Psionics play a huge role in that setting, and not having official rules for them has been a thorn in my side. I have the skills to work around the lack of official rules, but if they introduce official rules and they contradict what I've been doing, it will be a hassle to reconcile.
Why are psionics different? That is going to depend upon the campaign, but I can tell you how they've been run in mine and why I can't achieve the result I want by using the aberrant mind sorcerer or refluffling a wizard.
There are 5 types of magic in my setting. Three of them use the magical weave - Nature magic (which flows from the positive energy plane to the negative energy plane which flows to druids and rangers to form their magic), Divine Magic (which is pushed down through the weave to clerics and paladins from a divine power), and Arcane magic (which wizards, bards and sorcerers pull out of the flow of the weave with skill or force of will). Then there is supernatural magic, which covers all magic that exists in the world that does not originate in the weave, and includes curses, the supernatural, and science (like chemical combustion). Finally, there is Psionics, which also does not originate in the weave. Instead, a creature generates the power themself.
Psionics, in my setting, are not bound by the rules of common magic. Dispel Magic, Detect Magic, Counterspell, Antimagic Shell ... these spells do not interact with psionics because those spells interact with and attack the spell weave - and psionics have nothing to do with the spell weave. Psionics do not require spellcasting. They may manifest as ki for a monk, or power manifestations from a psion, or the thinly veiled force rip off I have for my psionic/psychic warriors. Regardless of how they manifest, they are a product of the psionic creature.
They originate, for the most part, from the Far Realm. In my setting, the Far Realm collided with the Known Universe 4,500 years ago. The collision shattered the Known Universe, creating my transitive planes (Shadowfell, Feywild, Ethereal), infecting creatures with chaos (turning Devils into Demons, introducing aberrations, driving Gods mad, creating 'dark heritages' (such as drow, duergar, gith, etc...), and began to introduce to the universe the unique magics of the Far Realm - psionics. They were essentially unknown in the Known Universe, but they were the only magic known in the Far Realm.
Historically, mechanically, Psionics have had distinct mechanics. Psionic power points were the primary mechanics in most editions. In addition, players interested in psionic characters have generally been the type that want to do funky stuff, so we've upgraded them. It has always had to adjust to the erratic way that TSR and WotC handled psionics, but we've always tried to strive to make psionics feel: 1.) Organic (as opposed to crafted), 2.) Dependent upon the inner strength of the Psionic character, and 3.) More 'Super Hero' than 'Wizard'.
We all know what inspiration we see for monks, and the same is true in my game. My psionic/psychic warriors are heavily inspired by the Jedi - players wanted it, and I provided it. Psions are your Super Hero and Horror Story figures, with clairsentience, metacreativity, psychokinesis, psychometabolism, psychoportation, telepathy, and metapsionics being the core of their powers.
I generally modified the systems provided so that psionics worked on a power point system and evoked the feel of comic book psionic battles. Psionic characters had a number of points, and could plug any number of them into a power. However, they rolled an ability check to make their power work, and the higher the power point investment, the harder it was to pull off - and failing could mess you up. However, Psionic PCs regenerated 1 power points by not using their powers, and the rate they recovered them depended upon how far down you were. If you uses a small fraction of them, they recovered fast. If you use all of them, it could take a long time to recover them. I also had a 'rock-paper-scissors' game worked out for the different psionic defenses and attack modes/forms. These were all about psionic battles, but they had a very distinct feel in the game, and tended to be very tense as the monsters and players revealed their defenses and saw how it impacted the environment. If you played the Intellect Fortress against Mind Thrust, the Thrust had an easier time penetrating the defense and being able to impact you - but it was a very effective defense against a Psychic Blast.
In the end, psionics were VERY distinctly different in feel than traditional spellcasters.