The biggest contrast is in demeanor. Claudio is an emotional character, but Benedick is a rational one. Claudio is caught up by the idea of love, swept away by his attraction to Hero. He does not consciously think of the idea of marriage and what it means - he just knows that he wants Hero. He speaks poetically about her, and Shakespeare portrays him as practically swooning in the first act.
But now I am returned and that war-thoughts
Have left their places vacant, in their rooms
Come thronging soft and delicate desires,
All prompting me how fair young Hero is.
If you have read Romeo and Juliet, think of Claudio as a slightly older, slightly more mature version of Romeo.
Benedick, however, is a man of wit. He constantly lectures on the drawbacks of marriage, explaining with intellectual consideration all the reasons he may appreciate women:
That a woman conceived me, I thank her; that she brought me up, I likewise give her most humble thanks
But he makes it clear that he will not marry as he considers it a type of slavery to "yoke" himself to a woman.
Another example of this contrast comes in the false accusation against Hero. Claudio, in jealousy and a fit of emotion, believes and acts upon what he has "seen." However, Benedick remains behind. He questions it, he considers the testimony of Hero, and he rationally concludes the truth.
However, although Claudio is less gullible than Benedick, he is also less stubborn. When he realizes that he was wrong, Claudio admits his mistake freely and performs repentance for it. He is willing to marry a cousin of Hero's as punishment for his "crime" and does not argue. Benedick, however, is not so easily moved. Although he has clearly demonstrated his love for Beatrice, announced it openly to her, he is quick to deny it in mixed company. He is backed up against his earlier protestations that he would never marry, and so doesn't want to admit that he was wrong. He actually says this to himself:
When I said I would die a bachelor, I did not think I should live till I were married.
He is showing us that he can't accept that he was at fault - he must change the situation to make it seem as if he was correct all along. Ultimately, as Claudio must admit that he was too quick to judge Hero, Benedick must admit that he was too stubborn, and they both are rewarded in the end.