I like psionics in game for a simple reason: The rules are simple to adjudicate!
Say you are introducing someone new to the game (heck, new to the industry) and they want to play some kind of 'spellcaster.' I use the term spellcaster loosely, as someone might who didn't know the difference between psionics and vancian magic.
Now, you can have these two approaches (assume core-type casters):
1. Well, you got your two choices. Wizards have to pay per spell, keep a spellbook, keep track of spellscribing costs, and then they have to meditate once per day and pick the spell they want to cast. Yeah, it's a bit of guesswork ... but you'll eventually figure out your pattern. Or, on the other hand - you could pick the sorcerer. It has a limit to the number of spells known and can cast them as many times per day as you have slots per level. But, yeah, the list you get to choose from is still pretty daunting.
2. See, this thing here called psionics means that you get a total. No such thing as spell slots per spell level. You get a total number of points. And, you get a certain number of powers known, like the sorcerer. The main difference is that the list of choices is also smaller, so you're less likely to make a rookie mistake than with the sorcerer. To use a power, simply subtract off how many points you use from the toal. That's it!
Now ... I'm not saying psionics is a better system than vancian magic (although I personally prefer it). But the thing is, psionics isa snap to teach to someone. That way, they can learn it and use it effectively on day one (assuming they can add and subtract, of course). It lets them get a feel for magic and how to use it. Once they get a feel for balancing a 'known powers list,' the combat system, and the rest of the rules of the game - then they can start to swallow the elephant known as "learning how to play a wizard well."
Do I allow psionics? Yes. I love it as a teaching tool. Absolutely love it as a teaching tool. Plus, I actually like the flavor of it better than wizard fluff. But that's just my opinion.
As a personal question: A few of the psionic lovers have mentioned Hyperconsciousness and a few other books. I have and love the XPH. I didn't like CPsi at all. What does Hyperconsciousness, Psychic Handbook, and other psionics style books have to offer ... and assuming I've got money to plunk down - what should I look at getting and why?