PNG is a really nice format to use while roughing in the design and CSS of a site. First, chop up your layout into 64-bit lossless PNG images, and build your CSS and HTML so everything looks pixel-perfect. The page size will be huge, but that's not an immediate concern. When all the design work is done, you'll optimize all those sliced up PNGs - some will be converted to JPG, others may become a smaller 32-bit or 16-bit PNG, or GIF - depending on its palette, transparency needs, etc. Some may even be discarded entirely, visually replaced by CSS techniques.

There are techniques using PNG that, if used intelligently, will save bandwidth - potentially lots of it! For instance, you can reuse PNG opaque-to-transparent gradients layering them over solid colors, to produce color gradients. If your design uses a lot of gradients as backgrounds with complex foregrounds, using PNG layering will prevent the use of images with large blocks of gradated colors (which are normally can not be compressed very well) behind foreground elements (which can be placed over the gradient as a separate layer, potentially with a much smaller palette size and antialiased transparency).

Images that are going to be shown over different background colors are a perfect fit for PNG. If you employ antialiasing in the alpha channel, your icon will never have that "jagged white halo"

The only true advantage of PNG is its alpha channel. I'm all for using PNG where it's appropriate, but alas designers who use PNG compulsively can get lazy about optimization (speaking from personal experience) and end up with huge 64-bit images consuming bandwidth where 64-bit images aren't needed. Using PNG is like cooking with salt. Use it sparingly where it's needed to enhance a flavour, but don't put it on everything. Some people put salt on everything. Yuck!

GIF has transparency too - selected pixels can be colored "see-through". In contrast, PNG can do has partial opacity - any pixel can be colored and semi-transparent. Adding an alpha (opacity) channel theoretically increases the file size by 1/3 (increasing from 3 channels to 4 is an increase of 33%). Choose your compression format conservatively - do you need opacity, or do you only need transparency?