There aren't really huge changes, but it's definitely more than a cosmetic update. As you noted, it does look different, but aside from appearance the smaller size makes it easier to fit into some HDMI ports that are a tight fit.
The bigger change is added support for dual-band WiFi 802.11ac and 5Ghz. Since one of the biggest issues many users had with the OG Chromecast was so-so streaming quality even with a fast internet connection, this should minimize the WiFi bottleneck that seems to have been the cause.
Personally, I have a 1st gen Chromecast and will probably pick up a news version eventually, but I'm not in any huge rush. If you have issues with congestion on your wireless network, though, it would make sense to go ahead and update to the new version.