Can you hook it up to a 120 watt amp?
Yes. You can hook almost anything up to almost anything else, with the right cord.
Would that blow the speakers?
Depends. Are you cranking the amp? If so, then it's possible. This cabinet was designed to go with a 60 watt amp, and probably couldn't take more than about 100 watts continuously when it was new. As the drivers age, some of the adhesives holding it together may have become brittle, and it could fail at high volume levels.
Will it blow up your amp?
Almost surely not.
Will the distortion get really bad?
Depends on what you think is bad. Personally, I think the sound of a dying driver is particularly nasty, so, if this happens, the distortion getting really bad (and the clean tone, even more so) is quite possible.
What noise should you look for?
Any significant change in the sound coming from the speaker without changing any of the controls on the amp.