The images look the same to me?
Any differences would be down to the 2 bits of software using different decompression & rendering engines.
Firefox (particularly web formats like jpeg) is optimised for speed, not "true colour" or high resolution (think many little pictures to make up a single page).
There are also standard "web colours" used by browsers where anything out of this colour band could be dropped (Its far fetched, but I would guess they apply it to the image rendering to optimise performance).
Photoshop also has its own decompression and rendering engine, however this one is designed for high resolution true colour images even in lossy (read shitty) formats like jpeg. Photoshop also applies its own custom filters/things to the image after the raw data has been loaded to try and improve the quality for use in the "lossless" format you work in on screen.
To get around the problem...
Try using a lossless format.. like png or bmp.
bmp files are just chunks of data, so there would be no decompression to screw with the colour but it would still be subject to rendering optimisations in browsers - but heavyweight formats like bmp tend to be less optimised due to the fact they are not used in pages.
Its also worth noting that I didn't research anything here... so you may wish to check these things out
The only person who i know that could maybe give you a better (fact based?) reply is shadow