I don't know about the picture quality question of your opening post, but obviously if you're being given $80/month google play credit you should definitely get something attached to your TV to access Google Play content.
By the way, how do you manage to get given $80/month credit?
Out of the Chromecast and Nexus Player I would get the Nexus Player. It's more expensive than Chromecast, but it has all the functionality of Chromecast and additional functionality that makes it far more appealing, IMO. It has a dedicated remote, an optional gaming remote, voice search, and it supports 3rd party apps and games which you may find a useful way of spending your credit.
In some respects it is a disappointment though. It does not have an ethernet port (so you are limited to wifi, like with Chromecast), it has a very small amount of on board storage, and I don't think you can playback content from sd cards - which I would have expected from an Android-based device. But the Android TV platform is open to other manufacturers so I would assume that other manufacturers would soon release their own versions of Android TV boxes that don't have these limitations. Therefore you might want to hold out until April to see if anything else comes onto the market.
A final option is to get a Roku, which has a Google Play movies channel. However, if you go down this route you don't get Google Play Music or Google Play apps and games on your TV.
Are you able to use your credit on hardware? If so, your first month of credit would almost cover the cost of the Nexus Player.