I've been thinking about getting these. Does the app do anything particularly useful (I.e. Any important features I'd be missing out on by not having it)?

As a side note. I can imagine why they'd want this information. Seeing what people are listening to and what volume settings are commonly being used would potentially help them tune future products better for their "average user"

The app allows you to use 1 headphone as a relay to another bose bluetooth headphone to play music from the same source, as well as firmware updates, renaming the device name the headphone appears as under bluetooth and finally changing the language of the voiceover (which announces things like device connections and low battery)

I concur. Have and love QC35 (got them on right now). I only downloaded the app to test some settings change, I think. It's not particularly useful for anything else. Since I don't have to use it to pair, I'm not sure how I'd know when it was "running" or not, short of manually killing it in the iOS task launcher/killer thing. Time to just uninstall.