'Chrome' is the Google branded officially released version of 'Chromium' - which, as you pointed out, is the open source base on which 'Chrome' built. There is currently no 'Chrome' release for Linux, only nightly builds of 'Chromium' - which can be had here, if you're using Ubuntu:
I understand that; what I haven't found is the actual difference between the two. Are there any functional differences between Chrome and Chromium (assume Windows version, since I'm not referring to things missing in an incomplete port), or is it only a branding thing so that the projects that fork and extend it don't use something with Google's brand?
http://ppa.launchpad.net/chromium-daily/ppa/ubuntu