I'll agree that they look interchangeable on paper, but in reality "Jeg vil gjerne bli kjent med deg" comes off as friendly and polite, while "I really want to get to know you" sounds somewhat desperate and may even come off as a little creepy when used in the same situations.
"Really" is a general intensifier, not strictly a positive one. You can think of "veldig" as the 1:1 general purpose translation of "very/really". "Gjerne" adds a layer of politeness and goodwill to the statement or request, which "very/really" lacks. That's why "gladly" is a better match in most cases, but often it's better to rewrite the sentence with "I would like/love to" as is done above.
Using "would like/love" instead of just plain "want" is a concept in English which doesn't translate directly to Norwegian, but has a similar effect to "gjerne" - which doesn't translate directly to English.
P.S.: "Veldig" can also modify "gjerne", so you can actually write "Jeg vil veldig gjerne bli kjent med deg", just like you could say "I would very much like to get to know you".