Ha! The never ending debate!!!!!
I could go on for pages on this one, disecting era by era for each.
I will leave this brief at first, and then maybe get more detailed later.
Here's the deal. Historically, I have always prefered Maiden and been more familiar with their entire catalog. Up until about 5 or 6 years ago, there had only been a couple Priest albums which I can honestly say I listened to and knew in their entirety (primarily Vengence and Defenders). To me, Priest always seemed a lot more simple and commercial driven than Maiden.
Over the past 5 years or so though, I have completely discovered Priest's ENTIRE catalog (aside from the Ripper albums, though I do know some select tracks from the 98 LIVE MELTDOWN double disc). My opinion of Priest has changed DRASTICALLY, as I have discovered MANY MANY amazing epic Priest tracks, such as Evening Star, Beyond The Realms of Death, Blood Red Skies, Touch of Evil, etc.
Overall though, I think it would be hard for me to ever definitively pick Priest over Maiden.
I have seen Halford solo, but finally saw Priest live for the first time ever (late comer to being a true Priest fan ,I know) on the Metal Masters tour a couple summers ago. Simply amazing. No other way to put it.
I will end now, since I have answered the initial question, with my pick being Maiden.
Maybe later I will go into more detail, as I have some issues with Priest I would like to discuss!