I would say they both mean pretty much the same thing: a mixed breed. I don't think either is necessarily pejorative at this point--maybe they were in the past. I've even heard people refer to themselves as a mutt, meaning they have ancestors from many different places.

From what I have read 'Mongrel' is more commonly used in the UK, whereas 'Mutt' might be used in the US or Canada. Although I have the impression mongrel can have a negative connotation. In South Africa 'mutt' is commonly used, some people might call a street dog a 'mongrel' here. I think in many places people opt for the term 'mixed-breed' nowadays.

Neither mutt nor mongrel are inherently pejorative, but both can be. Mutt’ is specifically related to dogs, and can be an informal word for ‘dog’. ‘Mongrel’ is rarer and more formal and clinical, and used for animals other than dogs, too. I’m going to give the mutt a walk. (Said affectionately) Some people prefer mutts to purebreds because they believe them to be healthier.

Well, I don't take even a penny out of you, but I will give you the most detailed answer as far as possible. Mongrel and mutt dog simply mean the same whereas mongrel and mutt both are offensive when referring to a person, mongrel means crossbreed while mutt means fool.

One is a real term; the latter is a pejorative. Google it.