If I have an image which is
2000px by 2000px, and the same image at
500px x 500px, what are the advantages of editing the larger image and scaling down afterwards?
You will not get the same result at all. When you start with your 2000x2000 pixel image you have some scene or subject in it. If you crop it down to 500x500 pixels then you will only have 1/16th of the subject in it. However if you have a 500x500 with the same but lower resolution content, you will have an entire subject.
What you can do instead is scale down the 2000x2000 pixel image to preserve its contents after you are done editing. This requires a little more computation power but any recent machine will handle it without problems. What it gives you is more precision. When doing a contour for background removal for example, it is better to do it at the higher resolution. The scaled down image will look better after.
On the other hand, some editing steps are better done in the final size. The most common example is sharpening because scaling does resampling over sharpened pixels, it will introduce artifacts. So the suggestion is to always do sharpening at as final step. Also something to do after scaling down but before sharpening is moire removal.
As you see, the reality is not as clear-cut as your question suggests.
If you mean scaling: You should do most of your edits on the original image size. That way you edit once and resize/reuse the final result for print (large size) and viewing on the screen (smaller size). If you do the edits on your 500x500px, you need to do it all over again for print and again in few years when resolution of displays increases and you will want to display at higher than 500x500 resolution.
If you really mean cropping: For some editing it is beneficial to have extra image area. For example, if you are fixing the horizon or perspective, even if you are cleaning some dust or unwanted elements near the edges.
There are several types of "editing". For example:
1) Just adjusting contrast, brightnes, etc. Advantages on doing it on a larger file. None.
2) Sharpening must be done on the final image, for example on your 500x500px one.
3) If you need to get detail from an image, for example take a specific sample on some point of the image to make the white balance, you need the best file you can get.
4) Photo editing as graphic designer. To cut some object from the background and paste it to another background you need the bigest file you can get, so when you sample it down the possible noticable errors and borders gets difussed.
5) You want a bigger edited file... edit the original, not the resampled.
So it depends.