Donkeys only come in one species: Equus assinus. They have a variety of sub-types within this species, including mules, miniature donkeys, mammoth donkeys and burros. However, though there are variations in appearance and size, a donkey is just a donkey. The differences between a miniature donkey and a burro comes in the usage and origin of the terms, not in the animal itself.
The word "donkey" is an English term for the small, horse-like animal we have seen on farms across the world. The word "burro" comes from the Spanish word for donkey, "borrico."
The burro is just a donkey, but called by a Spanish name. Usually these are the miniature donkeys that originated in Africa and came to the Americas with the conquistadors.
Domestic And Wild
The miniature donkey is a domesticated animal. So is the burro, but often a burro has become feral, meaning it was once domesticated but now lives in the wild.
According to the American Donkey and Mule Society, "the term [burro] is correct only when applied to the mid-sized types of donkeys, and more correctly only those who are wild in descent." Additionally, they note that it is not appropriate to use the word "burro" when referring to Miniature Mediterranean Donkeys, which are under 36 inches, or Mammoth Asses, which are over 56 inches.
In Name Only
The difference, then, between a regular miniature donkey and a burro is only in the name. In the western United States, people often refer to donkeys as burros; however, this is simply a colloquial term used in that region and it does not indicate a different species or type of donkey.