Having favor allows people to go to the Underworld and to Fissure of Woe from the Temple of Ages. UW and Fissure have tough monsters and nice drops (if you're lucky). Those places are where you can also get more cosmetic armor.
Favor for a region is gained by winning in the Hall of Hereos.
Having the favor of gods for your country is extremely useful when you are questing. Kneel at a statue/shrine of one of the gods and it will pop out. Talk to it and you can receive 2 blessings (for a small amount of gold), 1 of which increases your percentage (2% all the way up to 10%) and the other gives you a health regen of +2 for as long as you are in that area (i.e. not warping to another location).
Lord Garofoli it doesn't really have to be a guild, it can be a random group, but yes, it's a PvP match.
And yeah, Kuku Monk is right, a lot of people don't notice that shrines in EAs give you bonuses when you have favor. Most people just notice that you can get to FoW and UW, but going on quests is a lot easier when you get a cheap 10% Morale Boost and health regen.