I can't compare AT&T to Sprint, but I know that AT&T is slower in most places than Verizon. Verizon has more 4G LTE coverage. Can you double-check if your phone just says 4G in the Notification bar; or does it say 4G LTE? If it doesn't say 4G LTE, than it's not on LTE.
I switched from Verizon to AT&T about 2 months ago because I wanted the Galaxy Note. I knew AT&T was slower because I compared my HTC Thunderbolt's (on Verizon) speed against a co-workers Galaxy Note (AT&T). His speed results were consistently 1.xx mbps. My Thunderbolt's speed results were consistently 6+ mbps (17 mbps max). The test were run in our in Irvine, CA office. But I wanted the Note, so I had to go with AT&T.
When I was in the Detroit airport, my Galaxy Note was on 4G, not LTE. My phone was useless for web browsing. The web pages wouldn't even load. I couldn't even run a speed test!
However, when the phone is on 4G around Orange County, CA (it's on 4G speed a lot), it's slow, but the web pages load most of the time. AT&T's 4G speed seems comparable to Verizon's 3G speed.
I meet many people from all over the country - Kentucky, Indiana, Boston, Northern California, Minnesota, etc. Nobody has anything good to say about the AT&T network. They look at me like I was an idiot for switching to AT&T. (Everyone said Verizon's network has better coverage for voice and data). Yes, I didn't make a good decision because not only is AT&T's network lousy, the Note is the worst Android phone I have owned.
I wish I never left Verizon.