Translation:He speaks both English and French.
'both' tells you there's going to be two items listed so the 'as well as' becomes redundant. You're just repeating yourself, it sounds like saying 'he speaks English and and French'
You could use
He speaks both English and French
He speaks English as well as French
How would you differentiate between him speaking English and French to the same level of proficiency and him being able to speak English when he can also speak French.
Personally, in English I would use a comma for the latter: "He speaks English, as well as French", but I don't know which case "sowohl ... also auch" is used for.
It's clear that 'sowohl... als auch' is used in the case of speaking BOTH languages (without a comparison of proficiency). It is a bit ambiguous in English, but you might clear that ambiguity by saying "He speaks English as well as he speaks French" if you were looking for the comparison of proficiency. I believe in German there would be a different way entirely of saying this if you meant to draw a comparison of proficiency.
Not to get political, but I'm an active Couchsurfing host in the USA and I have NEVER had a French guest who spoke anything resembling good English (unlike the Germans, Austrians, Dutch, Belgians and Scandinavians, who are always fluent). Heck, even most Quebecois can't speak English.