To me, "I don't like gray" means the same as "I don't like the color gray". To me, they mean that generally speaking, gray is not one of my preferred colors.
But "I don't like the gray color" to me means that I'm looking at a particular thing and debating its merits (maybe a shirt or a couch or a painting) and I'm pointing out aspects of it that I don't like.
Does Esperanto similarly differentiate?
I tried to figure this out, and here's what I think: the description in PMEG:
says that when you use an adjective as a direct description of the nown ( which is what is happening here ), normally you put it in front of the nown, but it is also permissable to put it afterwards, particularly if you wish to add emphasis to the adjective instead of the nown.
So this is pretty much the same as in english. "Mi ne ŝatas la grizan koloron" = "I don't like the grey color". You're probably shopping for pants. The tint is incidental. "Mi ne ŝatas la koloron grizan" = "I don't like the color grey." That is probably emphasizing grey, as a color you don't care for in general because it is so depressing.
But the correct translation would depend on the larger context and is rather subtle. I think both sentences could mean either. There are shades of grey here.
Since word order can be changed drastically in Esperanto whilst retaining the meaning of the sentence, both can mean either. La griza koloro and la koloro griza mean the same, since gray is an adjective in both (-a). Perhaps it can be used for emphasis somehow, but grammatically, they do not differ in meaning.