The next step is WebGL2. OpenGL ES 3.0 adds a bunch of new features that are sorely needed for modern games and applications. For instance, it allows drawing to multiple render targets, which is very useful for virtual cameras in video games (although the original WebGL software could access this as an optional extension when supported). The addition of “Uniform Buffer Objects” is a better example. This allows you to store a bunch of data, like view transformation matrices, as a single buffer that can be bound to multiple applications, rather than binding them one at a time to every draw that needs them.
It's hard to describe, but demos speak a thousand words.
The news today is that Mozilla Nightly now ships with WebGL2 enabled by default. It was previously hidden, disabled by default, behind an option in the browser. This doesn't seem like a big deal, but one of the largest hurdles to WebGL2 is how the browsers actually implement it. The shading language in WebGL was simple enough that most browsers convert it to DirectX HLSL on Windows. This is said to have the added advantage of obfuscating the ability to write malicious code, since developers never directly writes what's executed. GLSL in OpenGL ES 3.0 is much more difficult. I'm not sure whether the browsers will begin to trust OpenGL ES 3.0 drivers directly, or if they finally updated the GLSL translator, but supported implementations means that something was fixed.
Unfortunately, OpenGL compute shaders are not supported in WebGL2. That said, the biggest hurdle is, again, to get WebGL2 working at all. From my talks with browser vendors over the last year or so, it sounds like features (including compute shaders) should start flying in easily once this hurdle is cleared. From there, GPGPU in a website should be much more straightforward.