By request, and this is a particularly good pair.
Blatant was coined by Edmund Spenser in "The Faerie Queen" in 1596, its etymology is questioned from two origins: first, from the Scottish "blaitant", an archaic form of the word "bleating" and second, from the Latin "blatire" for "to babble or blab". It has been argued that bleating is less what was intended than the blabbing. Bleating connotes an annoying sound, usually pleading and whining, while blabbing connotes a loud publication, telling secrets and gossiping. Without getting too much into blabbing, it just doesn't work once you get past 4th grade and learn that telling secrets is best done sotto voce. But it seems difficult to describe a monster ("the blatant beast") as telling secrets more than as making loud and annoying sounds, and therefore, I would argue that in coining the word, Spencer perhaps intended both elements, folding on each other with a good double meaning. But that doesn't help us, because now we have a word that since its meaning has not been reconciled, still has two distinct and unrelated prongs: (1) the bleating and (2) the blabbing. What is important to note, though, is that there is no element of "obviousness". Loud and annoying does not necessarily give rise to "obvious" (particularly since obvious relates first to things that are seen, not heard). Therefore, DD's Random House definitions which opine for some level of obviousness are just wrong as an over-extension of the word, whereas OED and DD's American Heritage definitions, which are just about loud and annoying are correct. So ultimately, the correct usage of blatant is solely the loud and annoying aspects. That there is a connotation of offensiveness appears to be latent from the blabbing origins (when you give secrets you are offending someone) and/or because of the extent of something that is really that loud or annoying could be offensive. Therefore, blatant as it might mean offensive may only apply when something is extremely loud and annoying (i.e., vulgar) or when being disrespectful.
Therefore, it is correct to say that as he got more drunk, his one-uping stories became more blatant. Here it can mean both the loud and offensive connotation, probably works best because of the double entendre. The blatant barking of guard dog brought the police. It's just loud and annoying, but not offensive, even if it were at 4:00am. It is a blatant mistake to wear red and green to a Hanukkah party. Well, color combination is loud, but here, it was intended because of the annoying and possibly offensive aspects as being disrespectful.
Meanwhile, flagrant comes from the Latin "flagrare" for "to burn", and has come to mean "conspicuously bad, offensive or reprehensible", the idea that when something is burning it is noticeable to everyone and probably not desirable, at least back before modern fire-fighting techniques were available. Remember, even signal fires gave away position to the enemy. Flagrant also still carries the meaning about fire and the quality of fire (red, hot), and has been used metaphorically in that context when describing emotions, such as desire, or war, but I'll focus on the "offensive" aspect, since this is a comparison to blatant. Here, flagrant means obvious, and since obviousness is an evaluative measure, it doesn't apply to people or things, but rather to ideas and activities. He was a flagrant fellow often investing poorly. When applied like this, it tends to have the tangible qualities of red or hot, which would be inappropriate, or else it sounds like it is a malapropism of frivolous, and should just mean wasteful. His flagrant "investing" in Ponzi schemes earned him a financial custodian. Better. The tell-all book about the his patient's psychology sessions was a flagrant breach of the psychotherapist/patient privilege. [Ed. note: they can't all be about attorneys...] Speeding and weaving on the Beltway are flagrant violations of the rules of driving. Flagrant seems to work best with intangibles. "Sampling" his mother's birthday cake the day before her 50th birthday party showed a flagrant disregard for her feelings.
It was flagrant lie to say that she had not been swimming when she showed up in a bikini dripping wet, but it was merely a blatant lie when she proclaimed that she had not enjoyed herself when we were also suffering from the heat. The second lie may have also been flagrant, but getting chlorine or salt out of hair is sometimes quite hard and the effort may not have been worth it, so we can't know for sure. As psychopaths substitute fiction for fact, what they think are merely blatant breaches of etiquette are actually flagrant disregards of social norms. Plaintiff's counsel's motion for summary judgment contained many flagrant misquotes from the deposition which became blatant to me the more he kept repeating them despite correction in the opposition, at oral argument and from the bench. [Yes, there must always be an attorney...]