'by consent' implies both sides get something in return for the consent. Too many examples to list. I do note it is a very useful tool for lawyers to use.
Essentially, a consent judgment means I cannot reagitate the complaint if I am found not guilty. By that I mean it differentiates from a discontinuance?
Discontinuance means same thing unless there is a right of reinstatement.
Otherwise if you do not like the offer, you risk it and the possibility of an adverse costs order, and adjourn the civil matter till after the criminal proceeding is over.