"Detest" and "despise" are not synonyms. To detest is to speak out against; to despise is to look down upon. Their roots come directly from Latin/French. If you detest something, you are issuing speech which indicates that you dislike it. If you despise something, you have observed it and disliked your observation (or, more usually, something about the detail of the observation!)
When people use these words in idioms like, "I despise and detest their actions," they are saying that they have seen the actions, they do not like what they have seen, and they are now speaking out to denounce those actions.