Basically, quantum computers (will) have a completely different processor architecture. They will be much faster (which will help them break standard encryption models used today, for example), but QC are built to better model naturally occurring phenomenons, which are generally not based on switches like on and off. Today's computers interpret these phenomenons at the software level not at the hardware level, like quantum computers will have a much better support for.
Regarding AI, QC, and consciousness, it's hard to say. It all depends on how we define consciousness, because there are many such definitions, and probably according to some of them, no non-human will be able to reach it. I don't have the same strict interpretation, so yeah, I think at some point AI would achieve consciousness. Could be only a few years away...
what would it look like to hook an AI up with quantum hardware?
I imagine at some point it would be as easy to use quantum hardware as it is now to use normal computers. But not in the near future, for the regular people, I imagine. As for AI giants, I presume they will train their models on QC as soon as the technology is available and stable.