I've worked with ASOS and AWOS-3, here are some observations . . .
With ASOS, the weather is transmitted automatically to the NWS and can be easily augmented by a weather observer (these are at a lot of FAA control towers). It is pretty good at figuring out cloud cover and precipitation. It knew when SPECIs had to be taken (and issued them automatically).
The AWOS system I worked with was not nearly as good. It did not transmit automatically to the NWS (but I think it can be set up to do that). It could not be augmented manually, so weather observations had to be taken manually and transmitted manually. It was poor at determining cloud heights and cover, and so-so at figuring out whether or not it was raining. It did not know when a SPECI needed to be issued.
Both had the ability to record a voice message (when the tower was closed, usually).
ASOS had the option of 1-minute weather updates or hourly updates when transmitting the weather (the automated voice on the phone or on the ASOS freq), so usually it is set for hourly when the tower is open, and 1 minute when the tower is closed (or when there is no tower). The AWOS system I used didn't have that option (as far as I knew).
At airports with control towers, the ATIS weather is usually pulled directly off the AWOS/ASOS screen while the controller is doing the recording. ASOS is tied into the NAS, so controllers can look up the METAR at nearby airports via the Flight Data computer if the airport is ASOS equipped. It's possible for AWOS-equipped airports, but someone has the manually enter the weather and transmit it to FSS or the NWS.