What benefit .png over .jpg?

[joe at actionline.com]

What (if anything) is the benefit of .png image files over .jpg image files?

I can't see any difference in image quality and .png files are 10 times
(or more) bigger than .jpg files.

Also, what is the reason or benefit of having hundreds of thumbnail .png
images in the /home/username/.thumbnails directory?  Is there any reason
not to delete all those?