Well, it's the main brain for everything; TV, Internet, Wireless internet (if you choose it). MTS Ultimate TV won't work without the 2Wire RG. Basically, it has a modem in it that receives the VDSL2 signal from the DSLAM, and routes it to the set-tops and computer(s).
The VIP1216 and VIP1200 set-top's use a technology called IPTV. I don't know much about it, but I just picture them like little computers, taking an internet-like signal and streaming it to a display device.
You could reuse your old router if you wanted to, it's just that MTS doesn't provide support for third party routers. It might be more beneficial in some cases if the customer's router is an "N" type, or if you need a lot of hardwired ethernet connections. With just the 2wire in place, there are only 3 ports for internet, but let's say you have 4 computers, you could run your router as a switch. Then I think you'd have 5 available ethernet ports.
Oh, and to get routers to work with the 2Wire box, you would have to reconfigure it, which is not the installers responsibility.