> Does this mean we'll soon have mobile web apps that rival native apps in terms of performance or will DOM manipulation still slow everything down?

asm.js has nothing to do with the DOM.So if the browser engine is slow, a fast javascript engine won't make any difference. It only make a difference with pure javascript computation as asmjs basically recreate an "assembly" on top of javascript, which allows to execute compiled C/C++ code.

JS is already plenty fast, it's just the DOM manipulation that screws it up. But asm.js with direct access to a <canvas> tag, WebGL or not, would probably be interesting.

The latter. asm.js is interesting if a part of your app is computationally intensive. For example, if you want to create ZIP files in the browser asm.js is awesome.