"Consent" means to give permission or to approve. For example, "I consented to the surgery." "Agree" means to be in unison or concord in a statement or an opinion. For example. "The panel of doctors agreed that the surgery should be performed." Full example with both "consent" and "agree": "The panel of doctors had a meeting. They came to a conclusion and agreed that the surgery should be performed. They explained the procedure in detail to me and listed the pros and cons. The following week I consented to having the surgery done."

Cosenting to something is giving your permission, usually in writing, but not necessarily. To agree, is just to be of one mind with the other person. If person A says ' I loved that movie ' to agree person B would say ' Me too, it was great ! '