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.
Here is the discussion of boast versus brag in Webster's Dictionary of Synonyms (1942):
boast, v. Boast, brag, vaunt, crow, gasconade agree in meaning to give vent in speech to one's pride in oneself (or one's family, one's connections, one's race, one's accomplishments, or the like). ... Boast is the general term; it may or may not carry a suggestion of contempt, or impute exaggeration, ostentation, vaingloriousness, or the like, to the boaster; [examples omitted]. Brag is more colloquial than boast, and carries a stronger implication of exaggeration and conceit; it often also implies glorying in one's superiority, or in what one can do as well as in what one is, or has, or has done. [Examples omitted.]
S.I. Hayakawa, Choose the Right Word (1968) puts boast and brag in a group with crow, gloat, pride, strut, and vaunt:
These words refer to feelings of self-congratulation. Boast may suggest justifiable self-satisfaction: a college that boasts an unusually high number of distinguished alumni.More oftn, however, the word suggests a self-important and tasteless pointing out of one's own successes [examples omitted]. Occasionally the word can refer to self-congratulation for a victory not yet won: He boasted that he would finish off the challenger in the first round. ...
Brag intensifies the note of tastelessness in boast, suggesting limitless conceit and, possibly, inaccuracy of the claims being made: [examples omitted].
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:
BOAST often suggests ostentation and exaggeration [example omitted], but it may imply a claiming with proper and justifiable pride [example omitted]. BRAG suggests crudity and artlessness in glorifying oneself [example omitted].
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.