(Well I suppose there’s a possibility he/she is both of the above, but that seems overly complex, especially if the people who wear the hat and cloak are not working together.)
So which is it?
Here’s what we know:
“Professor Quirrell reacted just the way you expected.” to Blaise Zabini.
Zabini thinks hat-and-cloak wants to cause trouble between Quirrell and Dumbledore.
Hat-and-cloak at least pretends to be a plotting Slytherin type who nonetheless cares about students such as Zabini. This fits two characters: the defense professor and Snape. It probably fits the defense professor better.
“(fleeting disorientation)” right before Hermione sees Hat-and-cloak for the first time in Chapter 79. Likely obliviation.
“What’s your name?”
The black cloak rotated slightly, back and forth, it didn’t look like shoulders shrugging, but it conveyed a shrug. “That is the riddle, young Ravenclaw. Until you solve it, you may call me whatever you wish.”
Hat-and-cloak’s name is a “riddle”. Could this be Tom Riddle? i.e. Voldemort? i.e. Quirrell?
Hat-and-cloak claims “I know the true reason for the coldness you see in Harry Potter’s eyes. I know the true nature of Professor Quirrell’s mysterious illness. I know who Dumbledore truly fears.” Of course, he could be lying about any of this.
Hat-and-cloak wants Hermione to distrust Snape, Dumbledore, and perhaps Harry Potter. Hat-and-cloak claims to be frightened of Potter.
Hat-and-cloak predicts that Lucius will come after Hermione.
Hermione recognizes the face of hat-and-cloak and is shocked by it. (Big clue.) On the other hand, is there anyone she’d recognize who wouldn’t shock her here? Then Hermione sees “…a shining lady, her long white dress billowing about her as though in invisible winds; neither her hands nor her feet were visible, her face hidden beneath a white veil; and she was glowing all over, not like a ghost, not transparent, just surrounded by soft white light.” but this comes after the recognition and does not appear to be who she saw first. I.e. there’s another bit of “(fleeting disorientation)” that probably means obliviation and memory charm.
Extra-textual: per author’s note it is indeed hat-and-cloak who obliviates Hermione.
Keep in mind that any or all of these could be lies or deliberate misdirections.
If it’s not the defense professor, I don’t know who it is. There’s just not enough evidence. Could it be the defense professor? Well, we never see them together. Perhaps I’ve missed a clue, but this seems to still be an open question. Looking around the Interwebs, there is no consensus on this point. Almost everyone from Dumbledore to Quirrell to a time-travelling Harry or Hermione herself has been proposed as the mysterious figure under the Hat-and-Cloak.
One final note: it is not totally obvious to me that Hat-and-Cloak is the one who set up Hermione, though it certainly could be. I just wouldn’t jump to that conclusion immediately. As with Quirrellmort, it is inobvious what Hat-and-cloak is actually trying to accomplish and what he/she really wants. Temporal order does not imply causality, especially for a one-off event.
The obvious and to me most likely solution, is that Hat-and-Cloak is indeed the defense professor, and what Hat-and-Cloak wants is what the defense professor wants: a magical dictator to take over Britain. But a final answer will have to wait for more chapters. We may just have to leave this as an open question until he/she shows up again.