Commitment is a word we hear frequently in our modern culture (see graph of frequency of use since 1800). Commitment is the trait of sincere and steadfast fixity of purpose. It is the act of binding oneself to a course of action.
Integrity (from "integer" = the whole of anything, a whole number not a fraction - think "whole character" not a fraction of one!) speaks of the unimpaired state of one's mind and heart, of moral soundness and purity, of incorruptness, of uprightness, of honesty. Just as we would talk about a whole number, so also we can talk about a whole person who is undivided. A person of integrity is living rightly, not divided, nor being a different person in different circumstances. A person of integrity is the same person in private that he or she is in public.