Two pros and two cons
Pro one: a mini is smaller than a standard bulldog, shorter and more compact, which means easier travel: smaller crates, smaller carriers, less weight to carry. A lot easier in cars and on planes.
Pro two: a smaller dog has less hair, so it's easier to groom, with less maintenance.
Con one is breeding ethics. A lot of mini English bulldog breeders focus only on size, not health, temperament and functionality like they should. Be very careful where you get a mini from.
Con two: the good old-fashioned bulldog snoring. The better the quality of bulldog, the less snoring you'll get, but you'll most likely still get some. Be prepared.