They mean the same thing, but there could be very nuanced differences in use. “Problem solved” is more of an exclamation because it omits the verb “is” and just uses the noun + predicate adjective. It is something a person might say immediately upon solving a problem, with a sense of excitement or boasting. “The problem is solved” is a complete sentence, and it sounds more like a neutral statement of a fact rather than an excited utterance.

"Problem solved!" is an exclamation of completion. It's another way to say "I did it!" or "We did it!" (both of which are excited exclamations) if what "we did" was solve a problem together. I often use this for very silly things. Like... I fix my mom's phone just by pressing one or two buttons. "There! Problem solved."