시시하다: trivial, insignificant, uninteresting, underwhelming, banal.
진부하다: banal, trite, hackneyed, lacking new idea.
Although they share the sense of "uninteresting due to a lack of freshness", the two are quite different in their usage.
시시하다 is a common everyday word that children use a lot but not so much the learned. Its driving sense is "uninteresting".
진부하다 is a more brainy hanja-based word used mostly in critiquing of artistic expressions to note a "lack of fresh new idea".
고루하다 is another word similar to 진부하다 but with an emphasis on being outdated and old-fashioned, so as to be stuffy.