This has to do with how PNG compression works.
Unlike JPEG, PNG is a lossless format, position-wise, but not necessarily color-wise.
That is - while JPEG stores transitioning of colors from one area to another, PNG stores the data pixel by pixel and then losslessly compresses this data.
The difference between GIF and PNG actually comes from traditional conventions -- GIF was traditionally saved as 8-bit whereas PNG, which came later, was often saved as 32-bit (true color + alpha/transparency). So, the 32-bit image will require four times as much uncompressed data per pixel, and depending on the specifics of the image, the more detailed the data, the worse it compresses losslessly.
Hence, if you were to compare a 8-bit PNG to a 8-bit GIF, the file sizes should be very similar. The same applies with true color PNG vs true color GIF. This explains why selecting the 8-bit option will solve the file size issue. Otherwise, it's effectively a compressed BMP.