when saved as a .jpg, the data for the image (the pixel map and such) are converted from one format to another. The Paint.Net files include several separate layers of data, and intricate formulas and protocols for handling how they stack. JPG just merges all of it into one layer of data, and often blurs edges, adds noise, and makes colors just a bit...off.
.PNG also only works in one layer, but it maintains pixels as they are, and doesn't seem to have a problem with edges or noise. As an added plus, it also supports transparency, both full and partial--if the viewing program (usually an internet browser in this case) supports it.
Well, it's a lot more technical than I made it sound, but that's about what it does. The multiple layers tend to take up more space, as does transparency. That's the short version of the filesize issue.