There can be honesty without integrity, but no integrity without honesty.
Although integrity needs honesty, it does not mean you always have to be absolutely honest to others.
It just means you have to be absolutely honest to your self.
Being honest to others is many times just giving a personal opinion. And that opinion might just as well have nothing to do with facts.
So integrity needs honesty, but honesty as an inner process. Looking at your own values and act upon them with respect for others.