Photoshop save for web optimization

Banding like this happens in reduced color bit depth, and it is cumulative.

You will always see some banding in an 8 bit image, but GIF is especially exposed because those 8 bit colors are re-encoded into a restricted palette of 256 individual color steps.

Every time you push 8 bit data through another 8 bit process, another layer of banding is imposed and added to the original. GIF encoding is one extra step. Your display is one more - most laptop/office displays actually work in 6 bit depth, and the extra two bits added by dithering.

You will get a slightly better result with PNG (not jpeg, aside from not supporting transparency it uses destructive compression which in itself adds a lot of banding).

The standard way to eliminate it with 8 bit data is to add a tiny amount of noise, which breaks up the banding. But again, you can't do that with GIF and the restricted palette. But you could do it with PNG.