I will speculate a little, though I do not know for certain.
Zooxanthellae certainly are responsible for a large percentage of photosynthetic coral colouration. It's right in the name - "xanth" comes from the Greek "xanthos," meaning golden or yellow. They produce a good deal of the golden brown colouration of many corals.
However, NPS corals generally don't have a lot of zooxanthellae...and are usually distinctly not brown.
Reds, oranges and yellows are some of the most common colours in NPS corals. Carotenoids are probably responsible for much of this colouration and definitely commonly found in corals.
Carotenes are one type of carotenoid. Most appear as varying shades of orange to our eyes. They are a photosynthetic pigment - they absorb blue light. Some carotenes are used as light-blockers - they can protect cells from too much near-UV light. I don't believe they are directly synthesized by any coral, but rather derived through diet. I wouldn't be surprised to learn that some zooxanthellae can produce them, though that's not directly germane to our discussion. These are also what give carrots and sweet potatoes their colour.
Another type of carotenoid is xanthophyll - basically a carotene that's been oxidized. These are generally yellow pigments...the yolk of an egg is coloured by xanthophylls. Also not synthesized, but very common in diatoms (and thus phytoplankton). They are also light-modulators.
Some xanthophylls and carotenes can be converted to retinol or retinal (Vitamin A), but I do not know if corals can create the enzyme required to do so. They are also great antioxidants.
Astaxanthin is another related pigment. Common in micro-algae, but not directly synthesized by corals, it can give pink, yellow, white and red colourations. It's the main component in giving farm raised salmon and shrimp their pink hue.
Melanins, porphyrins, pterines, flavonoids are also commonly found in coral. Melanins are certainly familiar to anyone who's been sunburnt - they do a good job protecting from overexposure to light and can impart browns and blacks.
Porphyrins and pterines are commonly purple and green. Flavonoids are yellow, as the name would suggest.
The last few I'm not particularly familiar with - we need a biochemist to wander over here and tell us more about them (and probably correct some of my mistakes with the carotenoids)!
Hope that helps!
David
EDIT: Beyond the dietarily-derived pigments, there is perhaps a genetic component to colouration. While corals that brood their offspring likely "seed" them with their own pigments before release, the broadcast spawners often produce offspring with very similar colouration to the parent corals. It's hard to rule out the environmental factors (again, diet) since we don't have a whole lot of data points with broadcast spawning corals, but if any of the pigments can be synthesized by the coral it seems likely a genetic component would come into play.
Also, do remember that many of these corals can be found right next to Acropora species, in full sun and shallow water. Just because they do not have zooxanthellae does not mean they are restricted to low-light zones - UV blocking pigments can be critical for zooxanthellate and azooxanthellate corals alike.
It also may be helpful to add that the term "pigments," which I used without explanation, refers to a chemical compound that reflects and absorbs certain portions of the light spectrum (which makes them coloured to our eyes). The portion of light they reflect is what we see them as - if a compound reflects only blue light, it will appear blue to us.
One final thought as well, on the short growing bushy Scleronephthya spp. - I have found that certain colour forms seem a good deal easier to keep than others. I've had specimens come in off the same shipment, same size, same growth form...but the yellow-orange ones seem to regularly fair far better in captivity (at least initially, I've never kept them truly long-term) than the purple or red individuals. If this is a common observation, it seems logical that the pigmentation is playing a much more important role than previously thought.
Carotenes certainly contribute to photosynthesis in terrestrial plants, though as energy transfer agents...not as principal conversion agents. It seems worth exploring if their role is somehow altered in coral. It would certainly explain why yellow-orange bushy Scleros fair better initially if they can gain even a small percentage of their energy needs from photosynthesis rather than strictly relying on what they can catch.
