WebGL + asm.js on desktop computers should be roughly equivalent to native iOS / Android applications on top-of-the-line mobile devices (with possibility to get better effects on desktop GPUs).
This is btw what's going on with these very fast Emscripten ports - they can be done so quickly because the bulk of the hardest work was already done before, when engine and game developers made their code and maybe even more importantly their assets work well on mobile devices.
If you noticed many of these fast Emscripten ports were not desktop ports but had already existing mobile versions before - Unreal Engine Citadel demo, Unigine Crypt demo, and now Monster Madness.
Nevertheless it's still very impressive, big kudos to Mozilla folks.
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[1] For example these iOS 3d games should be feasible on the web (runtime performance wise, big asset downloads may be pain): The Walking Dead, The Wolf Among Us, Grand Theft Auto 3 / Vice City, Infinity Blade series, Max Payne 1, Mirror's Edge, XCom: Enemy Unknown, Real Racing, Asphalt series, mobile Need for Speed series, etc.
Plus of course anything 2D - Angry Birds, World of Goo, Plants vs Zombies, Osmos, Limbo, etc.
Given that asm.js games tend to run at something like 15-30% of the speed of native code, I doubt we'll see Battlefield 4 or Starcraft 2 anytime soon. Additionally, given that the WoW game client is many gigabytes in size, I doubt we'll see that either, even if it could run.
So: Minecraft, yes. The rest? No.
But as they've had a team port (a trimmed version of) Minecraft to xbox live arcade, and the iPad, it would surprise me if they're not already working on a *.js version.
This is from a link in the article for more information. These Monster Madness fellows found Firefox gave them 33% of native performance, and Chrome 20%. Presumably IE would give you fractions of these figures.
I figured that, rather than citing figures from benchmarks, I'd go for the real world scenario that this release has given us.