Its shortcoming is the difficulty and annoyance when hand coding (without compiling it from another language). The browser is used more by encouraging the adoption of web standards through discouraging the usage of web standards. You can see where the politics can enter.
Still, it makes for great demos such as the cloth physics applet from James Long of Mozilla or, more amazingly, Unreal Engine 3. The upcoming UE4 is expected to be officially supported by Epic Games on asm.js (and obviously WebGL will be necessary too) but, of course, Epic will not prevent UE3 licensees from doing their own leg-work.
NomNom Games, a group within Trendy Entertainment (Trendy is known for Dungeon Defenders), became the first company to release a commercial 3D title on these standards. Monster Madness, powered by Unreal Engine 3, runs in web browsers like Mozilla Firefox and Google Chrome without plugins (although it will fail-down to Flash 11.6 if your browser is unsupported for the web-native version). Monster Madness is a top-down cell shaded shoot'em-up.
You can play, for free, with an anonymous token here. You can also visit their website to learn more about the closed beta for registered accounts. It is natively supported on Firefox, Chrome, and Opera. I am not entirely sure why IE11 is not supported, now that Microsoft supports WebGL, but there is probably a customer support or performance reason for it.