Yeah, I'm experiencing very similar quirks between those two devices. Like you, I have both an iPhone 6 Plus and also an iPhone 5S, and while I love them both, I am constantly noticing the same issue with battery life, plus also a really nasty habit of my 6 Plus having App crashes (...ALOT!!) Has anyone else experienced this?
My 5S is really stable, and works perfectly even with the same Apps, so I know it's the phone and not the Apps themselves. One other phenomena about this I have observed is that it seems to happen most whenever I move my phone. I could be wrong, but I seem to remember something about movement sensors in the 6 Plus which has a feature where it auto-recognizes when you pick your phone up (for the purpose of auto-answering calls), so I suspect that turning this feature off might solve the problem. However, I have been scouring through the settings to see if I can find this setting to switch it off, but can't seem to find it. So if any one of you great iPhone peeps knows what this setting is called (and where to locate it) I would indeed be eternally grateful if any of you can help a sistah out with this info?
Many thanks in advance.