I hear a lot about how people want you to be "honest" with them, and their request often comes across plaintively as if they are unable to ascertain whether your wordsare honest and therefore they essentially want you to self-consciously 'rat yourself out'. I've always considered that sort of attitude to be rather lame. You can expect that most people are "fair weather" honest folks, very few (if any) are completely honest, and some are very dishonest. If it's in your self-interest to be honest, you're going to be honest, but you certainly don't merit any approval for that. The line in the sand is drawn, when your self-interest or your self-image is in conflict with the honesty of a given situation, such as when you are accused of stealing something. The answer is pretty darn clear, either you did or you didn't and the only truly honest answer is the correct answer as towards your actual actions, but for some people when the honest answer puts them in some sort of trouble, or an embarrassing situation, they give a dishonest answer.
From my viewpoint, an honest man, could in theory steal something, and not be caught nor be called account on it, and thereby justify to himself that he was still an honest man. Whereas, a man of integrity would never contemplate the act in the first place, or, if he was placed in a position in which he did steal, he would own up to it as Jean Valjean did upon breaking a window pane and stealing a loaf of bread for his sister's children that were starving. He paid the penalty of his crime and served his time. Later in the novel, when the wrong man is arrested and misidentified as Jean Valjean and sentenced to death, the real Jean Valjean dramatically returns and is willing to face the music despite now being a respectable man, mayor, and successful factory owner with attendant responsibilities and duties. Valjean cannot allow a man to be wrongly convicted of a crime that he didn't commit, no matter the consequences for himself. These can only be the actions of a man of the utmost integrity.
If one wants to have good friends, you want friends of integrity. For instance, an honestman might tell you that you look tired and worn out whereas a man of integrity would inquire as to whether you had been getting your proper rest, good sleep, or whether you were under any undue stress without invoking directly your looks. An honest man, might say to his wife, when asked how she looks in a particular dress, "it makes you look really fat, honey," and not receive a smile or thanks in return. Whereas a man of integrity would ask his wife to look at herself in the dress from different angles, demonstrating perhaps how a touch here or there might bring out her beauty more, and so forth.
Integrity is having the conviction, the courage, and the will to do the right thing even under the most trying circumstances. Honesty is truthfulness and straightforwardness but often honestly lacks the requisite reflection, experience, and wisdom that makes up integrity.