Recently, some users have had trouble sending audio from Spotify. The main cause of these troubles is Spotify silently updating itself. Here's what we suggest to get back on track:
1) Delete all copies of Spotify on your Mac by placing them in the Trash, then empty your Trash.
Make sure to check and see if you might have gotten multiple copies of Spotify. You should check the two following folders, by using the "Go to Folder..." command in Finder's Go menu:
/Applications
~/Applications
2) After you've deleted Spotify, restart your Mac and download the most up-to-date version of Spotify directly from their site:
http://www.spotify.com/us/download/mac/
3) This download will be a disk image (.dmg) file. Open it, and copy Spotify to the "Applications" folder inside your Home folder.
If no Applications folder exists in your Home folder, create one, then place Spotify inside.
4) Launch Spotify to make sure that it works. Be sure to click Open in the dialog Mac OS X may show:

5) Re-select Spotify in the Airfoil source selection menu, at the top of Airfoil's main window. Do this even if Spotify is already selected as your audio source, using the "Other Application..." option to re-select.

6) Once you've reselected Spotify, click the Transmit button next to your desired outputs. This should get Spotify's audio flowing to your desired outputs.
More Information
It may be useful to have more information about this issue. When an update for Spotify is available, Spotify automatically downloads it. It then quits itself and launches the update, all without any interaction from you, the user. This is non-standard behavior for Mac OS X, and can cause issues.
Additionally, when Spotify is unable to modify itself in the main /Applications folder -- for example, if the user is running a non-admin account -- then Spotify will install the update to the user's Home folder (in ~/Applications rather than /Applications). After that, every time the version in /Applications is launched, Spotify will quit itself and launch the version in ~/Applications. Again, this is non-standard behavior for Mac OS X, and can cause issues.