These are both good words, although most people here would have a hard time telling you the definition of egregious. The words are different, although they have a related "feeling." "Flagrant" is something bad that is done where everyone can see, on purpose. "A flagrant violation of the law" means that the person broke the law and wanted everyone to see them do it. It is the opposite of hiding a crime. "Egregious" is something that is so openly bad that it is shocking. For example, "There were some egregious errors in the legal proceedings." In this case, everyone can see the bad things, and they are shocking, but it was not always the intention of the person for people to see the errors.