Binaural audio recordings are usually made with a Binaural microphone that records sound in a similar configuration to how human ears work. Or they're just synthesized directly as a stereo signal (that's usually how those "binaural beats" music is made). They're nothing more than a stereo file, and the audio has just been recorded in a certain way that mimics the human ear and head. This is why the virtual surround sound from binaural recordings sounds so convincing, but only when wearing headphones.
Spatial audio is a more generalized term. Let's say you're filming an action movie and have a scene where a robot comes from foreground-right, kicks a car, and the car flies over the camera and makes a loud crashing sound behind the camera. You can't just stick a binaural microphone on set and record that because it's almost all CGI. There is no actual sound of a robot kicking a car to be recorded. Instead your foley artists and sound design team will record and modify dozens (even hundreds) of sound sources and combine them together in software that supports a sort of virtual 3d environment, and save it in a format capable of representing this (like Dolby Atmos).
You can then take that Dolby Atmos data and in realtime compute how to map your virtual sound-sources onto things like a 64-speaker Dolby Atmos array in an movie theater, or a 12-speaker home theater Dolby Atmos setup, or apply an HRTF to convincingly map that audio to 2 headphones. A Binaural audio signal is restricted to stereo and intended to be placed directly in the ears.
The most important thing about spatial audio in AirPods is that they have head tracking, and so sound will appear to be coming from the same places if you move your head around. It doesn't work nearly as well without that.
Spatial audio is a more generalized term. Let's say you're filming an action movie and have a scene where a robot comes from foreground-right, kicks a car, and the car flies over the camera and makes a loud crashing sound behind the camera. You can't just stick a binaural microphone on set and record that because it's almost all CGI. There is no actual sound of a robot kicking a car to be recorded. Instead your foley artists and sound design team will record and modify dozens (even hundreds) of sound sources and combine them together in software that supports a sort of virtual 3d environment, and save it in a format capable of representing this (like Dolby Atmos).
You can then take that Dolby Atmos data and in realtime compute how to map your virtual sound-sources onto things like a 64-speaker Dolby Atmos array in an movie theater, or a 12-speaker home theater Dolby Atmos setup, or apply an HRTF to convincingly map that audio to 2 headphones. A Binaural audio signal is restricted to stereo and intended to be placed directly in the ears.