Sometimes these words are used interchanged in just two consecutive sentences, therefore I don't expect there to be a big and obvious difference. Nevertheless, since people use the English language, words get connotations and that's what I'm asking for.
I recognised that "boast" is being used more often than "brag", see Google Ngram, dict.cc, and Google Search by itself for that.
Best Answer
Here is the discussion of boast versus brag in Webster's Dictionary of Synonyms (1942):
S.I. Hayakawa, Choose the Right Word (1968) puts boast and brag in a group with crow, gloat, pride, strut, and vaunt:
A usage note in Merriam-Webster's Eleventh Collegiate Dictionary (2003) suggests that the Merriam-Webster view of boast and brag has evolved somewhat since 1942:
The general sense I get from these efforts to distinguish between the two terms is that, although both may be used in some instances to describe unjustifiable self-promotion, boast occasionally appears in connection with some objectively valid basis for high self-regard, whereas brag is almost without exception used in the context of crude self-aggrandizement. Still, I think that most English speakers today consider them to be virtually interchangeable when used in connection with the self-referential utterances of people who suffer from unduly inflated self-regard.