This is a rather wide question.
The main difference between purebred and mix is that with a purebred you know what you'll get.
Also, mixed-bred is not a breed. You have the spectrum of characteristics met in the whole wide breed dog world and a bit more at your reach. Mixed-bred dogs vary from carefully considered litters to the Heinz 57 you pick from the rescue after a puppy mill raid or life on street not knowing anything but that it is not of any recognisable breed.
You can probably predict rather well how a Golden retriever - Labrador retriever mix or a Dutch shepherd x Belgian shepherd will turn out as they grow because the breeds are not that different in size, looks or temperament. But cross an akita with a collie and you could get almost anything. Or have that random mix of medium-sized and small dogs guessing if it'll grow to be 50 cm tall and weigh 17 kg or 32 cm and 7.5 kg and whether it will be too interested in hunting rats and other small animals as it could contain terrier or dachshund or be completely safe around your bunnies.
In some countries or associations only purebreds are allowed to enter all or some trials. In my country a mutt can do agility, obedience, rally obedience, dog dancing, and SAR and trial in them. All hunting trials, shows, schutzhund/IPO, and national search and tracking trials are off limits because the parent clubs do not accept mutts to enter. Mutts cannot title as champions nor compete for national or international championships. There are some sports that are not recognized by our umbrella organisation and are smaller and I am not aware of their stand. Like weigh pulling or frisbeeing.
Comparing any breed dog to any mixed-bred is unfair. Would you ask what is the difference between an akita and a Shetland sheepdog, well, you might but your question sort of implies that the question could also be if there is difference between an akita and a Sheltie. The could be differences between a mutt and a pure-bred or not, but you cannot know the answer until you know which breed and what kind of mixed-bred you are talking about and why you want to compare them.
As to why I have a pure-bred... I ran into this litter online, thought that if the puppy will grow up like its mother that could be a good dog for me. The breed was on my wish list and a very reasonable choice anyways for my situation so I was lucky in that way. We do not have that many abandoned dogs here and the pressure and propaganda to rescue is smaller. And there was this little chance that I got involved a sport not accepting mixed-breds.
I have been pondering the breed or origins of my next dog. It will likely be another purebred because the chance to get a mixed-bred with the specs I have is minuscule. I have some requirements which I doubt many mixes will fill. Unless it is the random chance that intercepts my life and drops a mystery pup into my hands like in some movie.