Interesting responses so far.
I ran a pbem around an offical Catholic superhuman team for a short time. My approach was pretty simple, that for all the individual failings, cockups, career builders, and idiots, the Church tries to do the right thing. Of course being large and powerful attracts lots of nasty attention.
My reponse was grab the Order of Malta (the Hospitalers under their latest brand name 8) and make a superhero team from it. Already plenty of fancy uniforms, spiffy traditions, shiny boots and legal right to declare war 8) [A benefit of being signatories to a whole whack load of treaties and keeping around, its difficult to prove that they aren't sovereign when 19th century powers treated them like they were]
Of course the main focus was protecting the Vatican and Rome from evil villians, but I was managing to try and include a little political scheming in too. 8)
As for my personal experiences playing religious heros, its the small things that count, that and using it to really fill in background.