If we have an iMac and an AppleTV, is it better to use the AppleTV to pull content from the Mac to play it, or is it better to start from the Mac and use AirPlay to push to the AppleTV?
There's no difference in quality, so it's purely a personal choice and depends on how your home network and systems are set-up.
I have a Windows PC running iTunes 24/7 and that has my entire library of music and video on it, so the Apple TV can pull whatever I want from it.
Sometimes I'll play content such as BBC iPlayer or Youtube videos from my phone, or from my laptop if I've downloaded a video I want to watch without importing it on the other system but it really makes no difference. It's just about convenience really.
The only thing that really makes a difference is network connection for the caching, wired ethernet is obviously a lot faster than WiFi.
In my system, pushing from Mac works more reliably. Pulling from ATV sometimes gets drop-outs, but pushing from Mac is nearly flawless. For this setup (push from Mac):
Downside: you can't listen to iTunes on your Mac and push different iTunes content to the ATV.
Upside: you can blast the music to your Mac speakers and the ATV at the same time, and walk around your house and hear the same music in different rooms (if they are in different rooms).
When I play music using the ATV interface, the sound quality is horrible. When I hit play in iTunes and route to ATV, it sounds fine.
I've done a blinded test on this, and it is horribly obvious.
I'm not going to downvote the answer that says there's no difference, since it might have something to do with which version of the ATV I have. But on my setup, there is definitely a difference.
Note: This is for streaming music from my iTunes library to the ATV. Much of my music is not on iTunes Match.