Yes. I forget my phone at home all the time. While I no longer get phone call alerts or message alerts, it does my pulse, stores it, records any exercise I do, records my steps, how often I stand up, and calculates the calories I've burned as part of its in-built technology. It also does a number of alerts without the phone - and plays music through headphones, without the phone.
You can't download music to it directly, but you can transfer music to the watch from your phone, and the music is there until you delete it. Stores lots of data, really - as much as my first home computer.
You can, for example, download a map to your watch and leave your phone at home - that's a reason it's so popular with joggers, runners and bicyclists. You can even download instructions to take a particular route, which Siri will announce to you as you bike or run, even if your phone is miles and miles away.
I don't have the cellular version, just the regular version. The only reason I'd get the cellular is if I really were running or jogging so frequently that I wanted more streaming music (it will stream from my Amazon music account only if I take my phone).