Visibility is an animatable property according to the spec, but transitions on visibility do not work gradually, as one might expect. Instead transitions on visibility delay hiding an element. On the other hand making an element visible works immediately. This is as it is defined by the spec (in the case of the default timing function) and as it is implemented in the browsers.
This also is a useful behavior, since in fact one can imagine various visual effects to hide an element. Fading out an element is just one kind of visual effect that is specified using opacity. Other visual effects might move away the element using e.g. the transform property, also see http://taccgl.org/blog/css-transition-visibility.html
It is often useful to combine the opacity transition with a visibility transition! Although opacity appears to do the right thing, fully transparent elements (with opacity:0) still receive mouse events. So e.g. links on an element that was faded out with an opacity transition alone, still respond to clicks (although not visible) and links behind the faded element do not work (although being visible through the faded element). See http://taccgl.org/blog/css-transition-opacity-for-fade-effects.html.
This strange behavior can be avoided by just using both transitions, the transition on visibility and the transition on opacity. Thereby the visibility property is used to disable mouse events for the element while opacity is used for the visual effect. However care must be taken not to hide the element while the visual effect is playing, which would otherwise not be visible. Here the special semantics of the visibility transition becomes handy. When hiding an element the element stays visible while playing the visual effect and is hidden afterwards. On the other hand when revealing an element, the visibility transition makes the element visible immediately, i.e. before playing the visual effect.