Apples are granted by staff and pollinators. If they feel the post is of quality and contributes to the forum, then they can give it an apple. We each have limited numbers of apples to give away each day so the post had better be darn special for me to give it an apple. Completely unrelated to thumbs up or down.
Thumbs up and down, pie, apples, and a bunch of other things I don't know what, all go into a magic formula that 'ranks' the thread. You can see 'best thread of the year' or best thread ever in the new view. It's automated, but I suspect PIE is a big part of it. Paul likes PIE. If you want a thread to get more attention, it can be highlighted by PIE, and it makes the formula take more notice of it. Giving away PIE slices to posts in the thread also raises it's rank somehow.
The 'rank' of the member, as far as I know, is unrelated to thumbs up or down. It's usually something earned by posting exceptional content or being helpful and welcoming to new members or other reasons. There is often a correlation between apple earning ability and promotion, but not always.
I tend not to look at thumbs up or down. I also set up my screen so I can't see the name of the poster. I like to enjoy the content of the post without any preconceived notions. There was another forum I use to frequent, it had the option to disable viewing the "approve"/"disapprove" rankings. I wonder we could have that as a PIE feature one day?
As for the notification, you can set up your account to tell you when you get a thumbs up (in your profile settings: http://www.permies.com/forums/email/preference) and currently the "top __ threads every__days you might be interested in" email tells you when you got thumbs up. You can also look up how many thumbs up you or someone else has gotten in the person's profile, as well as see who's gotten the most thumbs up in the http://www.permies.com/forums/user/leaderboard. But, there's no way to know who gave you that thumbs up, just like you don't know which random pollinator, steward, volunteer, or Paul or Burra gave you an apple.
Thumbs up, like R Ransom said, do go into the formulas for top posts, etc. But, I think more than anything, the thumbs up are a nice way to say "Thank You!" or "I like what you said there!" to a post. I love giving them out. Since I've got pie, I can give out two thumbs out per post...and I usually do when I like something. It's almost the same as a facebook "like." It doesn't really do much, but it's fun to give and get, at least for me .
I like that our forum has so many different ways to reward, inform, and encourage posters. Some may not care for the thumbs-up, and don't want to know about them. That's great! (I'd never realized someone wouldn't like that info--I hope there can be an option to "opt out" for people like R Ranson). Other's post more when they get thumbs-up, or like to know the background of who's posting. That's great, too! I like how Paul has so many ways to encourage more people to post more good stuff on here, so the knowledge can spread!
Thumbs up don't confer any automatic advantages, but they do make the writer feel that people read and liked what she had to say. I know it gives me a little boost and encourages me to keep wirting. Apples and pie carry more weight as signs of approval, but can be given only by certain people, whereas a thumbs up might have been given by any member.
Works at a residential alternative high school in the Himalayas SECMOL.org . "Back home" is Cape Cod, E Coast USA.