First, it's important to remember that Yelp's raison d'etre is reviews. All else is secondary.
Yelp's friend-like features, of which there are two, are both oriented toward the discovery of new reviews. In some sense, posting a review on Yelp is analogous to posting a status update on other services.
You have a "stream" or "feed" of reviews recently posted by persons you follow. This is a good way of winnowing down the main feed of all reviews.
There are two ways to get someone's reviews in your feed: you can follow them, or you can be their friend. I'll explain the differences.
When you follow someone's reviews (the Yelp term for "follower" is "fan"), the relationship is private, known only to you. No one, including that other person, knows that you are following him/her. This is an asymmetric, Twitter-like relationship, except it's anonymous as well. Each Yelper knows the total number of his/her fans, but cannot know who those people are. Consequently, no approval is required to follow someone. Just go to their profile and click the "follow" button. He/she will get a notification that they got one new follower, but that's it.
Yelp's "friends" are, as with other services, mutual, symmetric, and public. You request to be friends with the other person and he/she has to accept (or decline). Your profile lists links to your friends; these connections are public.
But being friends with someone has no effect, other than to add that person's reviews to your feed. It does not confer any additional visibility into their profile, or anything like that; and you can't restrict the visibility of parts of your profile to just your friends. It does not enable any more intimate connection, such as private messaging, over and above what you already have.
On G+ I had said:
In my own view, I've come to disregard the number of friends someone has on Facebook, but I'm still quite aware of the friends number on Yelp. It's almost a class thing. Someone with 500 "friends" on Yelp is qualitatively different from someone with 5.
And Tony asked, "Why? What is it about Yelp friends that means you are more impressed by a large number of them?"
One point I want to make up front is that anything I'm about to say is a generalization and that there are certainly plenty of exceptions. But the overall patterns are more or less true.
First, Yelp is primarily a local phenomenon, whereas Facebook has no locality biases built in. Sure, most people's circle of real-life friends tend to be rather local, but that's a sociological phenomenon, toward which Facebook is agnostic. Yelp, on the other hand, is specifically local to the cities where it operates. Of course, people do travel or move to other cities, accumulating connections in each; but by and large, people select their friends -- which, remember, are only interesting for their reviews -- based on their reviews of local businesses. The locality factor is important because it necessarily constrains the number of people available to be your "friends". (And we must also not forget that Facebook has orders of magnitude more users than Yelp does.)
Secondly: On Yelp, you don't have a "wall". The only thing you can do is write reviews and talk in the public forum (Talk threads). Whenever you post in either place, your post is "signed" with your "avatar" (an iconified version of your current profile pic) and your friends and reviews numbers. Every time you see someone's face, you also see their friends and reviews numbers! In the Talk threads in particular, this is interesting because the threads are completely public; any yelper from any city can (potentially) post in any thread. So you tend to see a pretty wide range of people. Contrast that with Facebook, where (a) wall threads are limited to circles (to use G+ term) defined by the owner of the wall, and (b) each post is "signed" with the author's name and face, but not their friends count.
So, ultimately, I think it's just about becoming sensitized to the oft-repeated stimulus.
