What is"Compensate for Scene-referred Profiles" and when should I use it? By default, it seems to be turned on in After Effects. I've read through the AE manual but even there they kind of throw up their hands in the air.
You should always leave it on. "Compensate for Scene-referred Profiles" allows the After Effects color management system to properly handle media with a scene-referred profile (which includes most real-world camera formats). Without this option enabled, Ae will handle them as if they are display-referred, leading to a slight gamma error. I assume the option is only there so that people could render legacy projects with identical results after Adobe added support for scene-referred profiles.