I've been an avid Chromecast user since it first came out, and even upgraded to the newest version last year. The Chromecast is still a solid device, and it is amazing for the price; but the Shield just works so much smoother.
Connections are instantaneous, and I never realized that fast-forwarding and rewinding in Plex was supposed to happen as soon as you hit the button... no more "click -10s, wait 5s, click it again" - it just snaps right to the point you tell it to. On top of that, the HDMI CEC support is top-notch, even turning off the TV after it reaches the sleep time for the device. The remote that you can get for free right now is also really well thought out, and the ability to plug headphones into the remote is just cool.
On top of it all, the GameStream support has been fun to play with. I'm looking forward to using it a bit more in the future, but has worked great for the brief time that I've put into it.
TL;DR, if you're in the market for a good set-top media box, NVIDIA Shield is awesome. If you want to save $, Chromecast is still awesome as well.