While honesty can be defined in negative terms as the absence of deceit, integrity has to be defined in positive terms. Integrity is an active adherence to principles, to values and, for Carter, to promises.
"Integrity also involves 'acting on what one discerns, even at personal cost'” (eNotes).
Honesty is certainly a virtue, but the demands of honesty are not necessarily as rigorous as those of integrity. For instance, a person might honestly sell a product that has not been thoroughly tested under the premise that the product has not been shown to be unsafe. However, to act with integrity a person would have to either (A) explicitly warn a consumer that the product is untested or (B) refrain from selling the product entirely if there was any possibility of harm for the consumer.
Note the difference here is between a necessary course of action (integrity) and a passive adherence to the truth (honesty).
"Carter’s first mark of 'living the integral life' is that of reflection: 'doing the hard work of discerning right from wrong'” (eNotes).
If honesty only requires that a person avoid telling lies or intentionally deceiving others, it leaves open many possibilities to act immorally. One can be honest and immoral at the same time, as the example above indicates (albeit somewhat mildly).
Honesty is defined by a person's relationship to truth and deception, but integrity is defined by a person's relationship to principles, to codes of conduct and/or to morality. Integrity stands as a more broadly demanding concept, morally, in comparison to honesty.
Depending on one's values, integrity may sometimes demand action (politically in form of protest or socially in the form of aiding those in need). Honesty, to the contrary, only demands that a person refrain from lying.