Blatant means "offensively noisy" and "brazenly obvious" (a blatant orchestra, a blatant lie). Flagrant means "shocking" and "disgraceful" (a flagrant criminal act, a flagrant oversight). Blatant stresses offensiveness and noisiness; flagrant emphasizes evil and wrongdoing. One who eats peas with his knife commits a blatant error. One who drives a car on the highway while drunk performs a flagrant act.
Dictionary of problem words and expressions. Harry Shaw. 1975.