The Note 3 will definitely be getting 4.4.3. We just don't know when yet. The entire update is intended to be a bug fix update. That doesn't mean there may not be a few new features added depending on the phone, but the update itself is intended for bug fixing. There are long lists of bug fixes included available of you do a search for it.
As far as battery life, T-Mobile users already got a stability update last month that significantly improved battery life as well as adding download booster, kid mode, and VoLTE. I never had bad battery life on kit kat but the TMO update increased my screen on time by 30-40%. So your provider should be able to get you those improvements as well unless the battery difference is solely Tmobile specific features, which seems unlikely. I know my signal strength and connection has not changed at all, so it is not a matter of that, which would be the biggest provider specific factor affecting battery.