I'm not really going to talk about purebred dogs because we all know what that means. But I am going to say that we have to remember that at some point in time many breeds now thought of as purebreds would have fallen into the designer breed category had that term existed. Why? Because they were 'created' by combining stock breeds to get the desired characteristics and bred until they became recognized as a breed and maybe even admitted into one of the larger kennel club organizations.
So how is this different than a mixed breed? Well in my opinion a mixed breed, though it may or may not happen accidentally, is the coupling of two breeds without any control in place. If my Boston Terrier were to mate with the Dalmatian next door to me, I think we would have created Bosmations or Dalostons! But that's a joke! Unless the coupling of multiple breeds is thoroughly controlled over several generations to create offspring that meet predetermined qualities, well then they are just mixed breeds.
And finally, what's a mutt? I think a mutt occurs when we have to guess, by examining the traits of a dog, what 'parts' he might be made up from. And if my Bosmations or Dalostons should happen to then mate with another mixed breed or mutt some characteristics would become so watered down as to become unrecognizable.
All of that said, does it matter whether your dog is a purebred, designer breed, mixed breed or mutt? Absolutely not. What matters is that you pick a dog whose size, personality, and grooming and exercise requirements fit your life. And that you love it half as much as it loves you. I say half as much because though we become the center of our pet's universe they are, unfortunately, a somewhat smaller part of our universe because of our many responsibilities. But if you own a pet or pets, be sure they are very high on your priority list. They deserve it and will give you so much in return it's truly immeasurable.