Usually "Assisted GPS" also means the ephemeris data is downloaded over the network instead of from the satellite itself. The ephemeris is data that tells the phone where the satellites are; that plus distance to the satellites tells the phone where it is. The satellite download is very low bandwidth and typically takes 45 seconds or more, so using the network saves time to first fix. On my phone, I can see the data connection every time I use the GPS.

This raises the question of whether an AGPS phone will get a fix without a network connection. Can it get the ephemeris data from the satellite as well as from the network? I'm not completely sure, but I suspect my N97 can't. I recently had a holiday in Ecuador, where I kept my phone offline to avoid roaming data charges, and I was never able to get a fix from its GPS. (I didn't care much because I also had a Garmin device, which worked fine.)

Other phones may vary. If you are relying on having phone GPS when the network is down, you should probably test it before hand.