Ambient Occlusion vs Environment Lighting? What is the difference between ambient occlusion vs environment lighting?

To me it looks like environmental lighting turns an image texture(or single color) into a hemi lamp source that shines from all angles but casts no shadows.

While Ambient occlusion turns a single color into a hemi lamp source that shines from all angles but casts some shadows.

Difference being: one has the ability to cast shadows and the other has the ability to use a texture as a lamp?

Actually, it looks like  if you use environmental lighting, you automatically also use ambient occlusion just with less control over its shadows.

Im getting a circular error in my brain.
Can-Not-Compute

The manual says the difference is


  The difference is that Environment lighting takes into account the “ambient” parameter of the material shading settings, which indicates the amount of ambient light/color that that material receives.


But ambient occlusion has a similar setting called "Factor".

But the factor settings effects ambient occlusion for all objects while
the "ambient" parameter effects just a single texture.

Is that it?

Here is a scene in Blender Render with AO on:


and here is the same scene with environmental lighting on:


So again, what is the difference?
I wouldn't think of Ambient Occlusion as part of your scene lighting, it's really a tool to art direct your image, and (if you output it to a render pass) for extra flexibility in the post production of your image. As Duarte already mentioned it is not physically accurate but is much faster to calculate so it can be helpful to increase the visual appeal of the render.

If you look at many artists' making of videos, you'll generally see that the AO pass is added on in post and not usually as part of the scene render. It's technically a localized shadow effect that gives you more control over the contrast in crevices and surfaces that are close to each other. 

I generally think of it as a tool to give you more control over the image, it's more of an art direction tool than a realistic lighting tool. I think it's more helpful to think of it that way. I know you can have AO enabled and combined into your raw render, but it's much more powerful as a post effect.