HTC ONE vs S3 Battery Life
I had the S3 on SPRINT before making the switch to the HTC ONE on T-MOBILE.
I switched for the better coverage in my area, so, it's not exactly an apples to apples comparison, but, perhaps similar to your situation switching from VZW to AT&T.
The better coverage makes all the difference in battery life, as I'm getting 3-4 hrs more life.
A co-worker has an otherwise identical S3 on VZW & never needs to charge up during the day. I had to plug-in after about 7 hrs when on SPRINT w/the S3 & ended up getting extended batteries to make it through the day.
So, depending on how you use your phone during fhe course of the day will be the biggest factor in determining whether you'll be able to make it through a day w/o looking for an outlet to plug-in to before making it home for the day.
If a typical (work) day consists of having multiple email accounts syncing/sending-receiving texts/voice calls & light-to-medium on-screen time, you should be able to make it all day. But, as w/any smartphone, if on-screen time is high w/heavy use (gaming/videos,etc) you'll want to have access to a wall outlet/external battery pack, or, perhaps an extended battery case such as the one that MOPHIE or i-BLASON offers for an added sense of security for battery life.
In summary, I believe battery life is better than the S3,but, w/o having both phones side by side on the same network to perform an identical battery life test, I can't give a definitive run time comparison, only an educated guess based on past experience, & I'd say YES, the HTC ONE has a better battery life w/estimated 20% improvement over the S3.