[mike00632]

What do you mean by, "However, unlike Hangouts, it just connects to your phone to send and receive messages,"? My Google Voice number is my primary number but not my cell-service number. Would messages.android.com send/receive SMS as my Google Voice number or my cellular carrier number?

[JoshTriplett]

Whichever number you have working through the Android Messages app on your phone. messages.android.com just connects (via Google) to the app on your phone, displays your messages from that, and lets you compose and send messages via that.