Disfranchise and disenfranchise mean the same: to deprive of rights or privileges. Disfranchise is the traditional form, but it has given way to disenfranchise over the last several decades, and the latter now prevails by a large margin. This is the case in all main varieties of English, and it is true despite the fact that several major dictionaries—including Oxford and American Heritage—still favor the shorter form.
Etymology and logic rarely guide English usage, but it is worth noting that there is a sound basis for the shift from disfranchise to disenfranchise. Enfranchise is the verb meaning (1) to set free, or (2) to give rights or privileges. Franchise bore these senses long ago, and thorough dictionaries still list them, but the word was never commonly used this way, and it wholly gave way to enfranchise for these senses from the 17th century on. So while disfranchise might make sense when we consider its roots, disenfranchise makes sense in modern English because it is the opposite of enfranchise, not of franchise, which is now primarily a noun.
There remain generations of living English speakers who grew up when disfranchise was more common, so disenfranchise still faces opposition. But the longer form has now all but pushed the shorter one out of the language (more completely in current searchable news and blog writing than in books, but books tend to lag a few years behind), so there’s no need to appease the few who continue to resist the change (except perhaps if you’re writing for a very old-fashioned professor or boss).
Ngram
Just for fun, here’s the ngram graphing occurrence of disfranchised and disenfranchised (using the participles reduces false positives from dictionaries, usage guides, and the like) in a large number of English-language books and periodicals published from 1908 to 2019: Â Â
The corresponding graphs for American and British English (which anyone can re-create by following the link and changing the settings) look very similar to this one.