Is there a difference between a contradiction and a paradox?

Yes, there is.

If, for instance, I uphold truthfulness but am in fact untruthful, and if I deny the discrepancy, there's contradiction.
If I do not deny the discrepancy, well, that's paradoxical - I am truthful about my untruthfulness.

In contradiction there's conflict, in paradox there isn't. We demand consistency, and we call things contradictory when they fail to meet the demand.
 
Paradox is manifest the moment we let go of the demand; when we realize that the world does not owe us fulfilment. It's really something extra, a cherry on top. Fulfilment is a kind of a grace.

Paradoxes then could be doors to wider, limitless understanding.

Living with contradictions feels terrible; we constantly struggle to make the world available to our limited, past-based thinking.
In living with paradoxes, we make ourselves available to the world.