JPEG and PNG are very different file formats; any given image that is smaller in one may not be smaller in another. And furthermore, their quality is not directly comparable.
For example, photographic content is very well represented in JPEG. The subdivision-of-blocks composed with pattern recognition makes for a format that does a very good job of discarding visual information in a way that human eyes do not easily notice. Of course, a highly-compressed JPEG may throw away too much information and show the blocks and instantly break the illusion of photographic reality, but used carefully, JPEG is fantastic for photos of the 'real world'.
And computer-generated content is very well represented in PNG. The lossless encoding is great for showing the straight lines of standard computer-generated displays, and naively-created gradients are replicated exactly with PNG. Had JPEG been used for either straight lines or naive gradients, the shortcomings would stand out instantly. Also, because PNG can be palette-based, it can very efficiently store images with only a few dozen colors.
So, pick the file format based on its use: JPEG for photos of reality or for very good approximations of reality, and PNG for computer-generated content.