"Short" and "long" bows are mostly a D&D invention. While bows of different lengths existed and were used by different cultures, in function they are identical: a bow is a bow. So, the distinction of a "short" bow and a "long" bow is not historically accurate. You don't use a different arrow for a bow that is 2 metres long compared to 1.3 metres long.

Where one would see a difference in arrows is in considering the draw weight of the bow (in gaming terms, how more STR would be required to use it). In order for arrows to clear the bow and fly straight, an arrow must be flexible enough so that it can flex around the bow (and therefore not deflect off), but not too soft so that it can correct its flight toward the target. A heavier bow requires a stiffer spine, which can be achieved through choosing arrow materials, the diameter of the shaft, the weight of the point, and so on. It's more scientific today with modern materials offering more personalisation, but similar principles would have been understand by archers and fletchers.

What this essentially means is that, in the context of your question, if someone has an arrow that was made for an archer using a 60lb longbow, it should perform adequately for someone using a 60lb short bow. There are some assumptions and generalisations, but arrows were generally compatible.

One significant difference that can be made relevant is the use of "heavy" versus "light" arrows. Bows and archery styles that focused more on long range flight shooting (such as Turkish archery) make use of light arrows to go further. However, they are not as efficient in transferring energy, so have less penetration. In contrast, a heavier arrow will have more penetration, but will not fly as far. This difference is also seen in the style of fletchings used: flight arrows tend to have narrower fletches for less drag, while arrows for shorter distance have wider fletches for quicker correction but more drag.

As for interesting arrows, that's a very broad question and something you should look up yourself. There are a multitude of arrow points - whistling arrows, blunt or crescent arrows for hunting birds, chisel points for penetrating plate armour, needle bodkins for soft armour, and so on. I probably should dig through sources and examples, but as I said, this is very broad and it could take all day to bring up interesting examples that you could find yourself with some searching.

Edit:

I should also add that in gaming, "short" bows are typically portrayed as the light bow for close distance shooting and "long" bows are the heavier one with more power and range. This is not historically accurate either, so if you're trying to apply historical and realistic concepts into your D&D game, it might not fit the established meta and balance.

I will disagree somewhat with /u/nusensei on this, as he seems to be going with Jim Bradbury's definition of a shortbow as a composite recurve bow, rather than the traditional definition of it as a small self bow.

I've previously written on the topic of wooden shortbows and their use. In addition to the shortbows I mention in that post, which are military bows based on their context and dimensions, two shortbows from civilian contexts have been found: one 11th century shortbow from Pineuilh in southern France, and the 12th century shortbow from Burg Elmendorf in Germany.

To address your first question, the main difference between arrows for a longbow and a shortbow would be the length. Both the shorter draw length and the lower efficiency of shortbows mean that their arrows can theoretically have a smaller diameter shaft for a given spine, but the examples I've found aren't much different in diameter (8-9mm) from arrows we know belong to longbows (for instance the Nydam arrows). The other difference is that, being shorter, shortbow arrows will be somewhat lighter, which reduces the energy a little.