There is no significant difference in the subject matter tested, both assess your competence regarding scrum principles and methodology.
The difference is in knowledge vs understanding & practice:
PSM I focuses on your knowledge of the material, the test itself mainly consists of multiple-choice Q&A to test what you know about scrum.
PSM II on the other hand, focuses more on the practice and real-world situations: The test itself
has multiple essay questions, how you would respond in such and such situations. PSM II is also considered a more advanced level exam and it is harder to take.
As Majaii's answers states, the essay questions were removed from PSM II (now there are some in PSM III). Nevertheless, the statement that PSM II focuses more on the practice of the principles of SCRUM is still true.