They’re both good, but Last.fm wins.  How can this be? Last.fm’s amazing powers are put on display after the break.

Last.fm can perform these amazing tricks:

Tag (and Multitag) radio.  If you want to listen to ‘piano’ music on Pandora, you have to select a song or artist that exemplifies ‘piano’, then vote up/down songs until Pandora figures out what you want.  On Last.fm, you can simply tell it to play a radio station made up of music that has the ‘piano’ tag.  Multitag stations are particularly powerful because you can further refine your selection, i.e. ‘piano’ and ‘happy’.

More variety.  After listening to similar radio stations on both Pandora and Last.fm for 8+ hours, I noticed a lot of repeated songs on Pandora, but no repeated songs on Last.fm.

Apps (and API).  Google’s vision of the future puts the browser at the center of the universe, so call me a luddite, but I still prefer desktop apps for a lot of things.  Last.fm has a spiffy Flash-based player, but there are also a lot of desktop apps that are more pleasant to use (more stable, nicer looking, etc).  Pandora, it appears, is an entirely closed system without a publicly available API.  While developers have still found ways to wrap the Pandora Flash widget in apps that add some features (like Growl notification), they leave a lot to be desired.

Monthly subscription.  Want to try Last.fm out for a month, no commitment?  Its easy to do.  Pandora, on the other hand, only offers their premium service in 1-year chunks.  It’s only $36, but I still prefer Last.fm’s bite-sized, 1 month for $3, offering.