Honesty and integrity are linked but while honesty relies heavily on facts and objective information or situations, integrity relies more on subjectivity and circumstances in which a person finds himself, knowing how to behave or react in order to retain integrity and good will and meet expectations.
To define honesty requires stripping away any mitigating factors which may persuade a person to do or say something not completely honest. Telling the truth makes a person honest but, for example, if you are hiding someone to protect him from someone or something evil (using the Holocaust as an example), honesty would require you to expose the person hiding whereas integrity would insist that you keep his secret and do not disclose his whereabouts.
In manners, honesty and integrity sometimes cause conflict when, for example, a situation demands that a person is polite to, say, someone he wholeheartedly dislikes. Honesty requires that the person analyzing his own honesty remains true to himself as well as to the situation. As a person who is concerned with honesty, it may seem best to ignore or overlook politeness because it may contradict his principles. However, real honesty is not a simple matter and a person would surely not "honestly" say that hurting the feelings of others is his intention and so, calling on his sense of decency - his integrity - the honest thing to do might be to show respect, be polite and prevent hurt. However, a person who is totally honest but who has no understanding of consequences will always over-simplify a situation and be forthright, thinking that he is above reproach, whereas he is not being honest with himself; he is only selfishly protecting himself above all else. This type of misguided honesty may even be responsible for wars!
Communication relies on honesty but honesty depends on integrity in order to flourish and foster relationships. Honesty and integrity are not about personal preferences, looking good, shifting blame or protecting oneself. Honesty requires honor and, in fact, they clearly stem from the same root word. Honor demands respect and respect means manners.
Explaining the differences between honor and integrity is therefore about interpretation and requires an acceptance of the subtle differences between, for example, sincerity and compassion.
Honesty: Always tell the truth, be consistent and conscientious, be sincere, lack deceit, behave correctly, be straightforward and speak clearly so that double meanings and misunderstandings are avoided.
Integrity: Always be fair, just, compassionate, non-judgmental, respectful, helpful, self-aware, follow one's own conscience, be trustworthy (not quite the same as telling the truth!), know the difference between right and wrong and uphold high moral values.