This is an interesting question. As a question you can either ask: Do you mind if I speak in English? -or- Would you mind if I speak in English? Both questions are essentially the same question with the difference being the initial question word (do/would) Asking with 'would' is more polite, but using 'do' is not rude or wrong, just not as polite. If you are using it in a statement, then you can use the negative form (don't) Please don't mind if I speak in English - this is technically a correct sentence, but not used. It's more so insinuating that you are a nuisance because you are speaking in English. You could say "I don't mind speaking in English" or "I don't mind speaking English" if someone is switching to your language but you are comfortable speaking in English or if someone is trying to communicate with you through google translate apps.
But for your question about what the differences are:
"Don't you mind me speaking about English?" you are using the word "about" this means that you don't want to speak in the language of English you just want to talk about the language of English in an educational fashion OR about English people from England.
"Don't you mind I speak English?" this insinuates that you want to communicate in the English language.
I hope this helps and clarifies!