I always assumed those sites get their data from weather.gov or something similar.

It seems accurate enough at least to pick out clothes for the day. The precipitation forecast sometimes is a bit overzealous later out in the week but seems pretty accurate for the next 24 hours or so.

Also, how does weather.gov compare to OpenWeatherMap.com?

I'm starting one of these "weather on an e-ink display" projects, and wondering which API to use.