Like everything else it depends...
What are you going to do with the image?
PNG stands for 'Portable Network Graphics' and is best suited for server side/web graphics. It supports up to 48 bit images I think (huge). It also does 8 bit (256 color) like a GIF would.
TIFF is somewhat proprietary and comes in more variations. All TIFF's support a variety of compression schemes like LZW, etc. Newer TIFF can also support JPG compression as well.
If the image is a JPG, why not just leave it at that? You are not going to improve it (except by image editing). What you got is what you got.
A 1 MB TIFF cooks into approximately the following file sizes:*
660K for a PNG-24 bit
227K for a JPG at 100%
113K for a JPG at 50%
*These figures are arbitrary and dependent on the source image. Your results may vary.
Here are two good websites to peek at:
http://www.scantips.com/basics09.html
http://www.libpng.org/pub/png/
FZ