Manual focus is often used in situations where you need very accurate focus on a very specific part of the image, for example, for macro photos where the depth of field is so small you can't even get the entire subject in focus.

Another situation where manual focus is used when auto focus just doesn't work, usually because you don't have enough light.

And a third common use is "focus trap", you pre-focus on a specific spot, switch to manual focus to avoid the camera changing focus and wait for your subject to arrive at that spot.

For fast moving subjects auto focus is actually much better because, with any good camera, auto focus is much faster than manual focus and the camera has a tracking mode that it used to follow moving subjects.

Auto-focus is better in most cases simply because manual focusing with an high-resolution camera is pretty difficult or time consuming (at least with a camera up to entry-level DSLR range, never had an high-end DLSR to try manual focus on)

note: the above does not apply to rangefinder style cameras, those are usually designed for manual focus