Now that YouTube supports WebM (actually, since it supports HTML5), it's easier than ever to download the videos - all you have to do is look at the page source (remember to enable HTML5 for YouTube first) and search for "videoPlayer.setAvailableFormat": you should see some links and next to it, the video quality is displayed, like so:
Once you find the link for the video quality you want to download (unfortunately 1080p is not available in WebM yet), simply paste it in a new browser tab, then save it to your desktop and rename it to something .webm (whatever name you want, but append the .webm extension).
Here is a link example for a 720p video:
To be able to play the video, you need a WebM capable video player. The easiest way is to install the latest Gstreamer packages and then simply use Totem ("Movie player"):
sudo add-apt-repository ppa:gstreamer-developers/ppa && sudo apt-get update && sudo apt-get upgrade
You will get an error when upgrading to the latest gstreamer packages but don't worry, it can be fixed by simply running the following command:
sudo apt-get -f install
Note: Don't worry, installing the latest Gstreamer packages from the Gstreamer dev PPA will not break the audio/video playback for any media players!
You will also need the gstreamer0.10-plugins-bad package to be able to play WebM videos:
sudo apt-get install gstreamer0.10-plugins-bad
Now the only thing left to do is set Totem as the file association for .webm files - you have to do this manually by right clicking a .webm file and then selecting Propreties > Open With and add Totem ("Movie Player"). That's it!