I can't speak to the Droid, but on other Android phones the primary benefit is speed increases from offloading the apps to the SD card, leaving more memory for application multitasking. The Cyanogen mod also includes some RAM and CPU tweaks to get a little more out of the phone that might not be there otherwise.

So with the decent speed of the Droid, I'd imagine that the biggest advantage is the apps2sd feature. It's definitely annoying how little space is devoted to apps on these phones, and it's easy to run out. After rooting, I don't run out any more.

Oh, and I forgot about the other nice feature of rooting (though I don't think it's possible on the Droid yet): wireless tethering! Basically I can turn my G1 into a MiFi wireless hotspot. Very handy when I'm on the road and want to use my netbook instead and there's no WiFi around. Too bad I can't get it to work with my ZuneHD