I've run and played in a good few play-by-post (and play-by-email) games over the years; in fact, I credit a couple of play-by-email games with getting me back into RPGs in my mid-late 20s when I was living in Japan and wanted to get back into the hobby, but didn't know how.
A couple of times I've been involved in something very long-running. I ran a Planescape campaign via a Yahoo Group for nearly three years, between around 2004-2007, and was part of several long-running games run by others (one is still alive and well after 11 years, and I still get the emails, although I'm not an active part of it anymore). Yet for the most part, I've discovered that play-by-post is basically a bit rubbish. I don't feel like I'm very good at running such games and the experience is often slow and dull. I don't blame anybody in particular for this, because I've always found that everybody involved approaches the project in good faith, but there's something about the format which makes it easy to get excited about initially, but equally as easy to run out of steam.
Also, let's face it, there's a clear hierarchy of methods for playing RPGs, all other things being equal. At the top is the regular face-to-face group. Next in line is the regular real-time internet group played via roll20 or whatever (and I'd probably include play-by-chat alongside it). After that comes the face-to-face one-shot, and after that the real-time internet one-shot. And then comes PBEM and PBP.
However, that's not to say that play-by-post is without merit. From my experience with the format, I'd say it has the following benefits (not including the practical benefit that it allows you to play a game very conveniently, from the comfort of your own home, in your underpants, while eating blocks of cheese).
1. Because it is slow and fiddly to do anything that involves rules, the "game" element of the RPG gets critically diminished when doing a PBP, but because it is text-based and there are no distractions (uncharitably, because it's not really all that much fun), the "role playing" element of the RPG comes to the fore. Shorn of conversation, jokes, scribbling things on bits of paper, eating, drinking, etc., the game feels "purer"; players tend to identify with their character more strongly and invest in the game world more completely as a general rule. That has its drawbacks (it's less fun), but also its advantages if you want to be a bit more serious about things (and view that as being fun).
2. The DM can almost make things up as he goes along, reacting to what the players do, because there's no real immediate time pressure.
This means that there are three types of game for which I think PBP would really shine:
The one-on-one game. A DM and one of his mates could really easily do something interesting and different with the one-on-one format. A big problem with PBP is waiting for people to post. One-on-one has no issue there - the game moves at its own pace - and if anything is even more flexible than a face-to-face conversation: through G+ or emails or whatever you could run a game more-or-less constantly as a kind of back-and-forth. It would by necessity have to be quite different from a group face-to-face session, but I could see this working really well for a mystery or investigation-based game.
The Wild Cards model. If a group of friends wanted to develop a shared world a la Zelazny, GRRM and Pat Cadigan, you could absolutely do it through a rotating DM-ship PBP. Every month (or whatever) a different DM takes over and you run a new campaign with different characters set in the same world, or something similar.
The Birthright/Diplomacy model. Each player is the king of some realm, or the boss of a mafia group, or the CEO of a company, or whatever. They are all competing with each other and somebody acts as the central DM. The value-added of PBP being that you simply can't run a game like that in face-to-face sessions very easily, because each individual's turn might take ages to run, and because of the potential for spying.