Spent a summer interning at a small local radio station that produced old-school dramas, which gave me a front-row seat to this very question. The biggest difference isn't just what's written, but what the script assumes the audience cannot see. A stage play script lives in a visual world; you can write 'CHARACTER A crosses to the window, looking pensive' and trust the set, the actor's movement, and the lighting will sell it. For radio, that line is useless silence. You have to translate it into sound: 'A floorboard CREAKS under slow footsteps. A WINDOW SASH rasps as it's pushed up. A distant TRAFFIC HUM filters in.' The script becomes a blueprint for building an entire landscape purely through auditory cues.
Dialogue also shifts its weight. On stage, a character can convey a lot through a silent glare or a physical gesture paired with a simple line. In radio, that glare doesn't exist unless another character reacts to it vocally ('Why are you looking at me like that?'). Exposition that would feel clunky on stage ('As you know, Doctor, the serum is in the safe') can be more gracefully woven into radio dialogue, because the listener is already primed to accept conversation as the primary source of information. The stage script is a director's and actor's guide; the radio drama script is a sound designer's and Foley artist's bible, with every non-dialogue element meticulously notated.