can powers of different primes be equal? I am reading about Hilbert's Grand Hotel and, more specifically, the proof that the Hotel can accommodate countably infinite buses of infinite passengers each using the prime powers method. If I understand correctly this method is premised on the assumption that powers of different primes can never be equal. What might be a proof for that?

No; a consequence of the fundamental theorem of arithmetic is that if $p^k=q^\ell$ for some primes $p,q$ and natural numbers $k,\ell > 0$, then $p=q$ and $k=\ell$.