I'm just wondering whether "Edge Slide" and "Vertex Slide" are just two expressions for the same function or whether they work differently. Of course, when I slide an edge this means that the vertices which are connected by the moving edge are moving, but maybe there's a difference in the effect on the moving vertices. In Blender 2.66, the "Mesh Tools" in the "Toolshelf" only shows an "Edge Slide" button whereas Blender 2.8 gives me an "Edge Slide" and a "Vertex Slide" function when searching for "Slide".
In 2.66 there is also a Vertex Slide in the Vertex Menu (CTRL+V) with the shortcut: SHIFT+V.
When using Vertex Slide with two (connected) Vertices selected, you are actually sliding two Vertices, which makes it possible to do something like this:
You can even (but that is very finnicky and I didn't manage to record that)) slide them in two different directions.
Edge Slide always slides the Edge/Edges/Edge loop. When using the 'newer' (from version 2.63 onwards, I think) hotkey G,G then when you have a Vertex selected you get Vertex Slide and when an Edge (or two connected Vertices, etc) is selected you enter Edge Slide.
An Edge Slide can also be achieved by sliding the Vertices one by one in the same direction and by the same amount.
Thanks, spikeyxxx! I wasn't aware of that difference since I've started learning Blender with version 2.71 and directly learned the shortcut "G + G" for sliding one or more vertices.
Neither was I, but your question made me curious ;)
I actually wanted to do a slide like in the picture above a few times, but didn't know about the SHIFT+V, so I ended up sliding the verts one by one. So thank you for asking the question!