I always found this a funny etymological thing. Going just by the words themselves, if you asked me to guess which was which, I'd say they should be reversed -- "vegetarian" seems to clearly imply "eats a diet made up of vegetables" whereas "vegan" is more ambiguous and seems to fit "mostly eats vegetables but also will eat some non-meat animal products".

But what I'm guessing happened is that the less-strict diet came first (at least in the modern era), so got the "vegetarian" moniker, and then we just needed another work for the strict no-animal-products diet so had to come up with "vegan".