the second one is more formal but they mean the same thing

@Peachey - "You didn't need to take a taxi," means a taxi ride didn't happen. That's wrong. In both sentences, the most likely scenario is that someone took a taxi, but it wasn't necessary. Gracie's answer is correct.

Hi Tahar, This is upper-intermediate grammar, and a typical question on a placement test. "You didn't need to take a taxi," means a taxi ride didn't happen. "You needn't have taken a taxi," means a taxi ride happened, but it was unnecessary. The "[modal] have [past participle]" pattern tells us about an imagined past. It didn't really happen.

The second sentence is more formal, and you won't really see it being used in modern day english

@ Migi - it makes sense; it's just not very common.