PNGs can actually be around the same size as GIFs. It just depends on what type of PNGs you use.
PNG can actually function in two different modes (there may be more, not sure). They can have a pallete that is limited to 256 colors just like a gif image. This produces lower quality pictures comparable to a gif with a similar compression level to a gif. On the other hand, PNGs have an added option to display true color or all the colors (whereas GIFs don't have this option). This method makes for a larger file size, but an accurate image.