No completely sure if I've done this right but here you go (using Photoshop CC).
Quantized to 256 colours with dithering:
Quantized to 256 colours without dithering:
Looks to me like you've done a great job because missile's looks just like Photoshop's 
 
		 
Hey man, thx for these pictures! :+
Almost on-par with Photoshop regarding the non-dithered version. Their colors
are a tiny bit better distributed. Well, I've only applied some quick 
optimation to my algorithm ever since it was ready to run for the first time. 
I'm pretty sure I can pull some more optimization. It seems they use a better 
weighting strategy, which is completely plausible given that my one isn't the 
most advanced one yet easy to compute (least-squares).
However, regarding their dithered version, it's better than my one. (If you 
can even see the difference. xD) But that's for a reason, since I know where 
I am losing ground here. 
Can you tell me what option you have selected for the dither type? Because the
pattern used is not a Bayer one, it's more of a random type. Is there a name
describing the pattern/method used? Their pattern looks interesting. I'm 
cooking up a similar one, which is half random and half a pattern (for 
efficiency reason), which should result in some pretty awesome results 
visually and should be fast to compute as well as usable for realtime 3d 
graphics, which is a bit tricky considering the pattern is of random nature. 
I am interested if Photoshop's pattern here can be used in animations without 
the pattern starting to flicker due to it's random nature. Anyhow, I will need 
a little more time to develop that one.
	
		
	
	
		
		
			... IIRC, Gimp has a sort of shitty quantizer because Adobe have patented a better one (they call it q encoding)
		
		
	 
Ohh, I have to agree with you given that Photoshop really delivers some very 
good results in comparison. Perhaps I should spent my one to the folks of 
Gimp. But who does color quantization, retro graphics etc. with Gimp, anyways? 
There are better tools, I guess.