In a nutshell, Dolby Atmos creates the effect of watching a movie at a cinema or listening to music at a live concert — the sound comes from all around you (center, left, right, above and behind) — and Spatial Audio adds another layer that makes you feel like you're in movie or moving around at the concert.
When listening to music over AirPods, you can turn your head to left or right and the vocals and instruments will become more or less prominent depending on where you're facing. If you're watching a movie on your iPhone or Apple TV, and a character is yelling on the right side of the screen, you can turn your head to the right and it'll sound like they are yelling right at you.
But if you have at least a 4.0 surround sound system at home (not from Apple), you can hear more details because it has more channels than a stereo mix.