Try using Export>Save for Web instead of the normal export dialog. It offers options to fix this that the standard export doesn’t.
Your problem is that many web browsers (and other applications too) are not able to display the same amount of colors that Photoshop and many other professional editing apps. So basically, you are exporting an image that has more color information stored in it than the browser you are using can display. The browser sort of freaks out, and tries to show what it can – leading to the washed-out look you are seeing.
When you export, make sure you check the “Convert to sRGB” box as shown in the attached screenshot. This will make Photoshop resample the image, using only the colors that are “Web safe”, ie. that your browser (and everyone else’s) are actually capable of displaying. You should wind up with an image that look *much* more like your original. Freaked me out the first time this happened to me 🙂
