Asynchronous play means that your opponent doesn't have to be around at the same time you are. You can take your turn, then later on your opponent will take his turn at his leisure. This is increasingly more important now that mobile games are a bigger and bigger phenomenon. You can take your turn in some game while waiting at a bus stop or whatever even though you wouldn't be able to play an entire (synchronous) game in that same situation. So it's not just a "feature" when a game can be played in this way, it's transformative in that it allows you to play games in different real-life situations than you otherwise could. It also means you can play 10 different sessions of the game against 10 different people, all taking their turns here and there at bus stops or other free moments here and there. Synchronous games can't do that.
Although I play some simple asynchronous word games, when it comes to "real" games I generally scoff at asynchronous stuff. People ask if Puzzle Strike or Yomi could support such a thing and they pretty much can't because they are both so INTERACTIVE. I tend to put as much interactivity into my games as I can: the more the better, and the less the worse. The games that most lend themselves to asynchronous play are the ones with so little interactivity that you can do a whole bunch of stuff on your turn and the opponent can't even react. That said, allowing asynchronous play is still a huge win in convenience and maybe these days people don't want a "better" and more interactive game, maybe they want a game they can actually get around to playing. Often, that means an asynchronous one. (And does asynchronous really have to mean worse anyway? See below.)
One day I thought, "Can my card game Codex be altered to support asynchronous play?" It's actually ludicrous to consider. The opponent can react about 9,000 times per turn to stuff you can do. It would be like playing MtG asynchronously, where the number of back and forths waiting to see if they other person does anything is so large that you'd give up on it before getting through a single turn, probably. That said, it would be such a big win if it were possible…somehow.
The thing that made me consider this at all is