"Integrity" vs. "honesty"—what's the difference?

Integrity is about conduct; honesty is about adherence to the facts. The person who without fail submits his timesheet every week, seeking clarification if unsure how to charge some time, is acting with integrity. He is probably also acting with honesty, ensuring that the numbers he reports are correct.

The Wikipedia page about integrity is useful here. It defines the word this way:

a concept of consistency of actions, values, methods, measures, principles, expectations, and outcomes. In ethics, integrity is regarded as the honesty and truthfulness or accuracy of one's actions. Integrity can be regarded as the opposite of hypocrisy, in that it regards internal consistency as a virtue, and suggests that parties holding apparently conflicting values should account for the discrepancy or alter their beliefs.

To summarize (and simplify a bit): honesty is about what you say; integrity is about what you do. In your example, yes, the employee demonstrates both honesty and integrity, but the act of being meticulous about reporting his hours (assuming that doing so is, in fact, consistent with his espoused value system) is an excellent example of integrity.

Now, the above definition is essentially an ethicist's version of integrity. If the employee lied on his time report, but in so doing he remained perfectly consistent with his own internally consistent value system, most ethicists would maintain that he was acting with integrity—though not with honesty, of course. On the other hand, as the Wikipedia article states,

In common public usage, people sometimes use the word "integrity" in reference to a single "absolute" morality rather than in reference to the assumptions of the value system in question. In an absolute context, the word "integrity" conveys no meaning between people with differing definitions of absolute morality, and becomes nothing more than a vague assertion of perceived political correctness or popularity, similar to using terms such as "good" or "ethical" in a moralistic context.

In this sense, the employee would most likely (barring any number of vague hypothetical situations involving the "greater good") be said to have neither honesty nor integrity if he lied on his time sheet, no matter how consistent his personal views on the matter.

Integrity compares your behaviour to your own rules and self-definition. If your behaviour is always consistent with your belief system, you are acting with integrity. (Some people would further impose that they have to approve of your belief system.) Honesty has to do with whether you follow some specific societal rules around lying and stealing. If your own internal rules were precisely those same societal rules, then you would act with integrity whenever you were honest, and be honest at all times in order to have integrity.

However, some people have extra rules, and some people don't support the "no lying no stealing" rules, or at least don't give them top priority. For example:

• Someone who tells a lie they feel is excusable (I am sure Grandpa will get better; the dog went to live in the country; no of course that dress does not make you look fat) is acting with integrity even though they are not being honest
• Someone who tells the truth rudely and offensively may defend their actions as "just being honest" but if their stated internal rules are to be kind and pleasant, they are not acting with integrity when they honestly hurt people's feelings
• Someone whose internal code puts loyalty to a profession, country, workteam, or "the greater good" ahead of everything else may lie in order to act with integrity (think spy, undercover cop, soldier etc)

I wouldn't call your colleague honest just for doing timesheets promptly and accurately. Conscientious maybe. If everyone else fakes their hours, and it's the social norm to do so, then it is probably integrity that keeps the timesheets accurate, and it shows honesty also. If everyone else doesn't care if the owners can invoice on time, as long as the paycheques keep coming, then it is probably integrity that keeps the timesheets prompt. Honesty doesn't really figure into that part. And if the norm is "please report the time we estimated, even if you actually spent longer or shorter", then insisting on reporting something different may actually undermine claims to either honesty or integrity.

My own personal opinion is that honesty is truthfulness, always laying the bare truth out there even if it is to the detriment of the speaker.

Integrity is essentially a clock work of action based on honesty.

For example a really callous, procrastinating person can still be honest but cannot have integrity.