Fetch saves more battery then Push absolutely, and I use the app because I dont care for Push(instant email) or Fetch(fetching your email on a timer instead of instant email)
I use the gmail app to manually check my email whenever I use the app, this is how I use my email, I dont need to get anything up to date instantly(depends how you use your email, your priorities in life etc, whether you need it for work or etc), I just go to the app every now and then and check manually(saves most battery)
So if you don't need email instantly, for your phone to be constantly checking for emails, constantly updating, then turning off Push saves alot of battery