It appear they are very similar in being student-centered but the students explore the gaps in knowledge regarding the problem based on their prior knowledge and experience; but they do more inquiry by doing research on the topic or issue to better understand it. See link below for more elaboration.
I think the article provided in the link below can help you differentiate problem-based and inquiry-based methods both in the philosophical bases and operational aspects.
I believe that inquiry-based and problem-based learning are one and the same thing as long as the problem used is not the kind you find at the end of textbook chapters. Problem-solving is more likely to promote inquiry when the problem is open-ended (has many possible solutions), real (or potentially real), relevant (to students or to their future professional practice), unstructured (purposefully missing information), and within your students' zone of proximal development, i.e., not too easy nor too difficult.