I checked weblio. They both mean a forest that hasn't been touched by humans, but the 原生林 definition has the added element of also not being damaged by things like wildfires.
Feel free to read them yourself.
http://www.weblio.jp/content/%E8%87%AA%E7%84%B6%E6%9E%97
http://www.weblio.jp/content/%E5%8E%9F%E7%94%9F%E6%9E%97
They are very similar though, and are listed in each others' "similar words" section of the pages.
I asked a couple of native speakers ...
自然林 is more commonly used and generally means a forest that is "natural" or in other words is not a "tree farm" or garden/park.
原生林 is usually not used in spoken language because it refers to "untouched" or very primitive/ancient forest ... usually used in literature and has a 神話的 (mythological) feeling about it.
Thank you! :)
Difference between 自然林 and 原生林
In english terms it seems like they mean the same thing, but I wonder if there's a difference between them in Japanese.