You can't get "JPEG quality" out of a GIF. Nor can you get GIF quality out of a JPEG, for that matter...
Anyway, did you, by any chance, happen to have dithering turned on, before? Go to File>Image Preview and experiment with the settings, Start with turning on dither.
Hi - yes dithering on - def makes a diference.
However, I cannot get the animated gif to look as good as an still (single image) exported gif/jpeg or any other format for that matter.
Without seeing the image/animation, I can't make specific comments.
However. Keep in mind that the GIF format has an indexed palette of 256 colors, total, one of which can be transparent. If you have a still image of 256 shades of red, and a still image of 256 shades of blue, they'll look fine as GIFs. If you try to combine them as an animated GIF, then the palette will need to be 128 shades each of red and blue, so neither of the images as frames in the animation will look as good as they do as stills.
Also, Fireworks doesn't always give you the maximum amount of colors. If you have an image with 1000 colors, and you export to GIF, you probably won't get 256 colors in your palette. Go to File>Image Preview and see what your palette is. If necessary, add colors from your image where you notice banding (with dithering off).
Finally, the two image formats are not equally appropriate for all image content. GIF is good for line art of solid colors. It isn't good for photos or gradients. If you can forgo your drop shadows you'll probably get better quality in the essential parts of the images. JPEGs are good for photos and gradients (although I prefer PNGs for gradients, myself), and are not good for line art. Unfortunately, if you want an animation, you need a GIF or Flash.