A horizontal expansion, like the name implies, is a expansion of the graph on its x-axis. The same can be said for the vertical compression, only that it's "shrinked" vertically if you will as opposed to "stretched" horizontally. To answer your first question, no, there is a difference beside these transformations being applied to different planes and also the visual changes.
In the function y = 2f (1/3x), the "2" on the outside will always double the values in which the y values will become otherwise. For example, if y=x and x=1, then y=x. But if y=2x and x=1, then y=2. What the "1/3" does on the inside to the x is that it triples the y values in order for it to equal the x. Normally in the equation y=x, both sides are equal, but if y=1/3x it also means that 3y=x. It all sounds complicated all right but i think you get the picture.
If a graph, say a circle, was horizontally expanded by a factor of 2, it would look "fat" or "bulged" to the left and right when all of its x values are doubled, but its y-values would remain the same. A vertical compression of a factor of 1/2 would then be different since it would halve all the y-values and effectively making the graph look more "squished" vertically than anything else, at the same time its x-values are not modified. Regardless of the factors, horizontal and vertical stretches are different because they only modify the x and y values, respectively.
In regards to the explanation of why the stretch factors of the y and x axis are stated apart from each other, all I can say is that since they are separate entities, it is not possible to state them in a combined statement. Normally, you would say something along the lines of "a vertical stretch of a factor of 5 along the line x=1, followed by a horizontal stretch of a factor of 2 alone the origin." due to the fact that there is no single transformation that could achieve the same effect, therefore promoting us to make a combined statement. Again, I feel that it may still be confusing for someone in your situation, but unfortunately that's just the way math works. It's not visually appealing by any means, yet if you can grasp the details in your mind, then it can certainly work for you.