The word 'psionics' has the same general meaning as the word 'bionics'. The term refers to enhancing the powers of the mind with a machine just as bionics refers to enhancing the powers of the body with a machine.
In a magical fantasy setting like D&D, true 'psionics' means something like an Ioun Stone - a magical means of enhancing the mental prowess of the bearer. A 'helm of telepathy' is a 'psionic device'. There is in my opinion room in the fantastic for both 'psionics' and 'bionics' so long as we are clear what they mean and use the term correctly.
The term 'psionics' comes to us from science fiction as a device which would allow its user to, by means of some sort of super science, emmulate effects which we would normally associate with 'magic', namely things like 'telepathy' and 'clairvoyance'. For example, a radio implant which communicated directly with the mind of the bearer and which allowed him to communicate directly with other such modified individuals would be a basic form of psionics. In this sense, the 'Borg' from Star Trek are a race of psionicists.
Over time, the original meaning of 'psionic' and its usage in providing a relatively hard science justification for the presence of seemingly magical effects was erroded in science fiction to the extent that while the clearly psionic Borg (which are not thought of as psionic) live along side magical races like the Vulcans and the Betans who have magical powers which are not justified by any sort of technology. They are simply 'empaths' or 'telepaths' without any explanation. That isn't psionic; that's simply magic.
In RPG's, 'psionics' has come to mean something which is in my opinion utter nonsense, namely, a form of magic which is not magic but which has all the attributes of magic including access to almost all the traditional real world magical abilities - telepathy, clairvoyance, precognition, etc. In the real world, you'll find very few examples of anyone who was claimed to be able to hurl balls of fire or conjure up walls of stone. But there was widespread belief in the arcane powers of the mind. In the real world, the idea of the 'psychic' appears in the post Enlightment/Industrial Revolution as a means of thinly disguising belief in magic behind a veneer of science. It is from this movement that we get scientific sounding descriptions of magic like 'telepathy' and 'clairvoyance' and 'extra sensory perception'. In my opinion, at some point RPGers got the notions of 'psionic' and 'psychic' irrevocably twisted together (as they are now in pop culture in general). What's made this worse is that there is an entire genera of fiction out there now which takes the superficial trappings of science fiction like ray guns and space travel and uses it is a vehical for telling very traditional fantasy stories complete with wizards and magic. Popular examples would include 'Star Wars' and 'Babylon 5'. These are not generally recognized as fantasy by the viewer, who has come to accept there conventions, and one of the tricks by which the story teller manages this is to use the world 'psionic' in place of magic where 'magic' would break the suspension of disbelief but 'telepath' passes by without question as a standard convention of the genera.
If you look at a D&D spell list, all the powers associated with psychics in the modern mind show up as spells. The presence of psionics is therefore not only superfluous, but in some cases down right inexplicable and reveals a fundamental dissonance and incoherence. At best 'psionics' is an alternative non-Vancian magical system either replacing or coexisting alongside the Vancian system. At worst, it simply reveals a lack of serious thought.
One of the more clever implementations I've seen is in GURPS. In GURPS, 'magic' is a system of magic designed to be roughly balanced with warswords and mail. 'Psionic' is the term in GURPS for a system of magic designed to be roughly balanced with access to assault rifles and ray guns. Hense, you can run both generas, but you can't necessarily successfully run magic and psionics in the game campaign (at least, not if you care about mechanical balance).