Flagrant and blatant are very similar in meaning as in doing something bad conspicuously./ in a obviously bad way Flagrant is stronger than blatant in that it attaches a sense of doing it willfully especially when knowing that is bad or it is it so obvious that it cannot escape attention/very noticeable His father is powerful , so he can behave flagrantly. The mistake he made is so flagrant that a simple guy can see it. He blatantly display his wealth ( show his wealth unashamedly)

I think most of us would treat these 2 words as interchangeable. If there is a difference, it would be a very subtle one !