I was going between these two phone as well and decided on the HD Maxx because of the battery life, here's why. I had a galaxy nexus with the extended 2100 mah battery, the same size battery thats in the GS3. I couldn't get through a day without having to throw it on the charger by 5pm and I could consider my use moderate over 3G (because I had to turn the 4g off) to save as much battery as I could. I'm a driver and the truck I drive doesn't have a place for a charger so I had to put it on the charger as soon as I got home. With the HD Maxx I haven't put it on the charger before I goto bed under 50% yet. This is with 4G on, screen brightness at 40%, GPS, sync, all of it on. Its great not to have to worry about battery life.
Of course there are some downsides to the phone, ROOT and ROMS are limited but I hope thats just because the phone is new, and the camera isn't the greatest are really the only things I miss from my Nuexus.