Hi,
I don't think you can go wrong with either.
The S3 has a big battery, not as big as the Maxx but it is removable.
Other features have the S3 better. I own a Maxx but I just pre-ordered my daughter an S3.
I had a final upgrade and we are all unlimited so I used it for her phone.
She didn't want the Maxx and they way she can burn through a battery, she needs something removable so she can carry a bag of spares. (Music playing, games going, screen on constantly, texting.) She is the torture test for battery life on any smartphone.