If the source of the stream adjusts for the type of device accessing the content, you could see less ISP data use.
For example, if Netflix streams 1080p max to an iPhone, but 4K to ATV, you might see less ISP data use watching 1080p content on the iPhone, whether or not you Airplay it to the ATV. The Airplay link is WiFi between ATV and iPhone, not subject to ISP data caps. I don't know if Netflix does this, just using it as an example.
But, if you have already downloaded content to iPhone (or PC or Mac), the airplay will be on your network only, not subject to ISP data limits.
That said, I have Comcast 1TB\Month service. I use ATV exclusively for TV viewing, including DirecTVNow, Netflix, Hulu, and a variety of streaming apps. The only TV viewing we do that is not streaming is Over The Air local signals for news etc, streamed locally on my network from a Tablo DVR connected to an antenna.
My data use is typically under 600Mbps, so no fear of going over. We work from home several days per month on top of the TV streaming.