https://www.dropbox.com/s/s3wlvb7h36o5uvb/20111022_131203.jp...
There are your two links. One is a public folder the other is not.
One takes you to a web browser before you can even get the link. One does not.
Guess which one I would rather have. And now guess which one they are removing :(
But clearly a) if it ends on .jpg it should be an image and not html and b) the direct link should be on the html page with a fancy "Copy to Clipboard" button next to it.
But instead of (b) they are actively preventing getting a link to the image by hijacking the context menu.
https://www.dropbox.com/s/s3wlvb7h36o5uvb/20111022_131203.jp...
Magic!
https://photos-5.dropbox.com/si/xl/Y-dAUouiiTV2IbOuAMZKe2t7k...
I am not sure if it could work at all times, but you can check for the URL of the picture within the new-style dropbox page.
Which is of course not same as Public folders.