There are probably a few different ways things could go wrong, but this was one that was highlighted to me recently.
Imagine you were uploading user avatars, and you saved them as their original name, you store all the avatars in a folder called avatar. I upload my avatar to my account and it's called andrew.jpg - if someone else uploaded another avatar, and their avatar was also called andrew.jpg, it would, likely either override my avatar, or fail to upload, but maybe create a DB entry with the filename as andrew.jpg, and then they would be using my avatar.
A trivial example, but if you could do the same with another more sensitive document, it could cause problems.
I had an app once where I stored the original filename in the DB as well as the hashName filename, and when it came to download the file, I would stream it from a controller with the original filename.
I did it like this, as they were worksheets for parents to download for kids, and they were named according to what type of worksheet it was.