Spotify Connect lets you control playback across your devices, and easily switch which device you're listening on. It's an enormously convenient feature. You can play music from your TV and control it with your phone. Or you can listen on your computer, but if you're getting ready to walk away, you can switch to your phone.
It would make Pandora an even more effective competitor if it had something similar.
More details here.
Moderator Edit: Changing title for clarity
This was a very important feature for me when I was on Spotify - it's so wonderful when you can use your phone or ipad to control playback on a larger sound system! And having joined Pandora recently, not being able to do this really nags at me - it's irritating to have to give that feature up (it's not the only one).
Allow users to transfer play session to another device. At times I am listening to a certain station/song from one of my devices at home (for ease of understanding Pandora via my cable tv settop box) and wish to pickup where I am/transfer the same station on the same song (and algorithm) to the next device (car/phone) should there be a need to more my session to a new device.
This is a critical feature that Pandora lacks. This is the biggest reason that I use Spotify for much of my listening.
If my wife is logged into our account on her phone and streaming music, I would like to be able to skip songs or change stations from my logged in phone without stopping my wife’s phone from actively streaming.