Research shows that a well recorded nature sound produces results close to the real thing in terms of its effect on the nervous system. The brain isn't especially good at telling whether a sound's source is real or recorded while it's processing it.
The real difference shows up in quality. A low quality recording with an obvious, repeating loop lets the mind notice it's artificial, and the calming effect drops.
So it's not about real nature versus a recording, it's about how natural and loop-free the recording feels. A good recording can carry much of what an actual forest gives you.