The biggest advantage for play-by-post for me is simply the the amount of time and effort dedicated to it is able to be spread out more evenly. The biggest disadvantage is the slow pacing involved in play-by-post.
Sitting down to play an Role-playing Game face to face at a table top or in a virtual space such as via a virtual table top tends to require several hours of time dedicated to play which requires that other facets of life be put on hold. Having a full time job along with various family obligations tends to fill up a schedule very quickly and it is often difficult to work in three to four straight hours of game time. In my case, doing this requires that my spouse take up child care responsibilities without any real assistance from me for at least three hours. Anyone with children (Especially multiple children) knows that this is asking a lot from ones spouse even if it is only done on a weekly or bi-weekly basis.
When it's all added up, I probably dedicate just as much time throughout the course of a week to my play-by-post games as I do to my table top games, but the difference is that the time spent on play-by-post is spread out over the entire week rather than all at one time on one day. So, while I may put in three hours to a play-by-post game (Or series of games) this translates to thirty minutes on Monday, maybe ten minutes on Tuesday, forty minutes on Wednesday and so on. I can take fifteen minutes during a break at work and post something. I can tuck the kids in at night, post something in a game and then go to bed or I can post something after dinner and check back later that night but overall I use up only about fifteen to thirty minutes. If I'm playing face to face or live online I am pretty much out of the picture for the majority of three to four hours, and this is definitely not something I can do on a regular basis.
On the down side to play-by-post, the pacing can be very slow at times. A single "Scene" that would take a few minutes in a live game may take a week or more in play-by-post, especially if the players and/or the game master have difficulty packing their post with as much information as possible resulting in several posts being made that could have been reduced to one or two.
Play-by-post I think seems to require a greater ability to think ahead and really be able to put as much information in a post as possible. During a live game a player can declare an action and the game master can ask questions regarding the player's overall intent or ask clarifying questions, but in play by post if the action or the intent is unclear it is going to require a post or several posts in order to communicate effectively between the player and the game master so that the game master knows exactly what the purpose of a roll was or what the player is hoping to accomplish with a particular action. The game slows way down as a single action is discussed. Play-by-post seems to require more prompting and recognition among the players and game master when prompting is taking place.
Players can take advantage of the pacing in play-by-post by expanding on their characters motivations and internal monologues in their post to add flavor, but if they don't do this then it seems that you end up with a series of one liners (I.e. Character does such and such then waits; little/no description of the action, just a declaration) and the pacing begins to feel slow and choppy.
I also think that when posts are not descriptive enough or they do not contain certain prompts it can further stall the game. For example, if a game master describes a scene but there is no catalyst included to prompt action from the players, the players may not know how to respond. In a live game they can simply ask the game master questions that can eventually work to prompt the character action but in play by post you once again are faced with a major pacing issue as the players attempt to elicit information from the game master (Unfortunately I think this is my primary short coming as a game master).
Play by post gives those involved a great opportunity to really think through a response and develop their character, as Thomaslee said, you can really go into detail about what your character is thinking, feeling, etc. On the other hand, as Malcombshaw suggested, the social interaction is not there as it is during live play. The players don't know a whole lot about each other aside from what they choose to share in their introductions.
Here is one advantage of Play By Post over table top, its easier to build relationships among characters. You might think the opposite is true but it isn't. In a Play By Post environment its easy to chose one of the characters in the party and seek out a romantic relationship or even vengeance without feeling uncomfortable as in being around a live table. If you did that you would get the look or the scorn, in Play By Post you don't have to worry about it.
That's true about the building relationships among characters. Unless you have a familiar connection with someone for real doing this in a live session might be a bit a weird or even offensive.
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Comments: If you do not like your local group and can't get to a tabletop group that's further from your location play by post and online tabletop are both great options. If you do have a fantastic live group then I'd prefer that to play by post or online but I still wouldn't give up either I just would not be as involved.
I enjoy both types of games. I have a tabletop group that I play with every 2 weeks. It's a great social event, and we all have similar interests apart from gaming that we participate in together. However, once every 2 weeks is about all the time I can devote to a 6 hour game session.
Play By Post allows for some play during the interim. While it is inherently slower, it provides for a little bit of fantasy and creativity every day. Knowing it is slower, selecting a system that is not combat centric, but more RP centric helps speed the plot along.
Play By Post also assists in improvement of creative writing, as Malcomshaw points out.
Unlike other game types that require an immediate answer when playing in a group playing by post or in a forum allow you lots and lots of time to think of a poignant in character response or action for your character. While some may use this to their advantage others will discount it as an annoyance to what should be a faster paced game. The latter should take up playing video games instead or find a tabletop group who will entertain their need for speed - in my humble opinion. .
Here's an advantage… Im writing this from my mobile in a park. I can use anything around me as a feeling for what my character might be doing. That can probably work really well for a Dungeon Master who needs inspiration.
I have been doing both for about 10 years now, and I would say I enjoy both equally, but for different reasons. For Play-by-Post you get so much more detail and richness in the game. If you can find good people to play with, then you can develop really deep characters that have real emotions and motivations. But it does take a long time to actually play through any individual scene. It also sucks in Play-by-Post when people get tired and drop out (Especially when it is unexpected and sudden).
However with Tabletop, you get to really see the character portrayed by others. Now again it depends on who you play with and how into it they get, but you can really hear and see how someone wants to portray a character. Also you get immediate feedback, so if something is unclear, you just ask and get your answer. If you attack some creature, you find out immediately if you hit or miss and what the consequences either way are. However with everything happening so fast, tabletop seems to be heavily combat focused and only 2D characters are created that have little motivation outside of finishing a quest for quest's sake.
They each have their pros and cons, but I personally enjoy Play-by-Play more for the depth and immersive nature of the game.