They are digital audio formats.
Very generally speaking
When people say adat, they refer to 8 independent channels of audio over a toslink/optical cable.
When they say spdif, they refer to 2 independent channels of audio over a digital coaxial cable.
This 'divide' is not strict though. Spdif can be carried over toslink, and there are converters to switch between carriers.
Spdif input is common on audio interfaces and some preamps and effects units will have spdif output , but is also often used to carry an encoded dolby digital format in home theatre systems and pc motherboards. This is then externally decoded into 6 analog channels of audio.
Adat is also common on interfaces nowadays and usually allows you to add an 8 channel of preamps or converters with adat output.
Sometimes it is used to carry four channels at a higher sample rate.