Boast, brag, vaunt, crow, gasconade mean to give vent in speech to one’s pride in oneself or something (as family, connections, race, or accomplishments) intimately connected with oneself.
Boast and vaunt are often used transitively as well as intransitively; the other words are chiefly intransitive.
Boast is the general term; it may or may not carry a suggestion of contempt or impute exaggeration, ostentation, or vaingloriousness to the boaster.
Brag is more forceful 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.
Vaunt is more literary than either of the preceding terms; it usually connotes more pomp and bombast than boast and less crudeness or naïveté than brag.
Crow usually implies exultant boasting or especially blatant bragging in a manner suggestive of the triumphal crowing of a cock.
Gasconade is the least common of these terms and implies habitual or extravagant self-vaunting.