For the user almost none. The difference is for the devs. If I want to make a cross platform game now, I have to worry about which Graphics api I will use, OpenGL, Direct3D, Vulkan etc. What sound api I will use. What Display, Input, abstraction I will use. Etc. If I could have a unified api and runtime abstracting all that already for me, making full cross platform games would be easier, assuming I don't want to use Unity. But that is of a much bigger scope that just WebAssembly. We already have that power, but we are chained to the browser. The discussion is to free that runtime from the scope of the browser. Decouple it.
The final dream is having a unified, Desktop, Consoles (Ps4 and Switch) and Browser runtime with a unified API (where sensible). One can dream.
One does not simply develop cross-browser. Different browsers have different issues with different graphics cards. As soon as you're not drawing a textured triangle anymore, things get hairy. Same for the audio API. And don't get me started on the sad state of WebRTC and websockets.
Yes, it's just not viable to develop for browsers unless you have the maintenance budget for a moving target. I decided on that after living through the Flash era and trying HTML5 for a little while. The browser is backwards compatible only for certain kinds of applications.
That's the thing. You don't escape from those problems in native environments. You still have to deal with different GFX drivers doing different crap on different platforms, cards that they that support a GL extension but don't, etc. And that's just OpenGL.
With the browser, you're just adding another abstraction layer that has even more complexity. WebGL is far for a decent spec. And we're talking just graphics, as I mentioned before, audio and networking have tons of unstable and untested code.
Multimedia programming is already hard on native environments. I don't see the web being a friendly environment anytime soon. There's no silver bullet, "code once, run anywhere" for games and multimedia.
The final dream is having a unified, Desktop, Consoles (Ps4 and Switch) and Browser runtime with a unified API (where sensible). One can dream.