Pandora's concept is to be able to determine what music sounds like, and find other music that sounds similar. Last.fm's concept is more social networks centered, using tags, recommendations, and listening habits to figure out what other music you might like based on what you're listening to.
One of the things they do right is you can get started right away without signing up or downloading anything. This is what drew me in in the first place. I could see the benefits of the service without having to sign up beforehand.
To get an idea of how easy this is, I typed in one of my favorite artists, "Nightwish", and it came up with a tag cloud that included "female fronted metal".
I clicked on that and it generated a radio station of songs that had all been tagged "female fronted metal". This is basically my favorite genre, so it's almost like having a custom radio station that plays mostly songs I like.
Here's an example of a widget the site created for me to embed wherever, that lets people listen to music based on my music preferences. Pretty cool, but apparently the music isn't always available to play due to radio broadcasting laws. Hrm.