First, let's establish that at least some male characters do have nipples in The Original Series :
Given what we see above, it's certainly not a general policy on TOS to have no nipples. With that out of the way, we can focus on the appearance of the Greek god Apollo.
Here he is in a larger, better screen capture:
There are nipples here; they are just especially pale and small.
Here is another, just for good measure — the nipple, while small, is clearly visible in the lower half of the image:
So now one can ask: are his nipples naturally pale and small or was something done to them? The answer is the latter. Commenting on the controversy over the idea of having the character of Carolyn discover that she is pregnant at the end of the episode, actor Michael Forest (Apollo) said:
I don’t know. It’s so long ago. I suppose it would have been OK, but you have to remember that in those days, the idea of that sort of thing happening in film, and certainly on television… you couldn’t even talk about it, really. To give you an idea, they had to put tape on my nipples for that episode. Can you imagine? They put tape on a guy, on a guy’s nipples. I mean, come on. But that’s what they did, and then they put makeup over that. I thought at the time, “I can’t believe what they’re doing.”
(Source — thanks @ThePopMachine)
So it seems that makeup tape had been placed over his nipple, and then his nipple was remodelled with makeup, possibly to do with taboos at the time, as implied by the actor himself.
One could ask why the same policy didn't apply to Sulu (or Kirk). Possibilities:
Perhaps the Apollo actor's nipples were quite pronounced and edging on "indecent" (by the standards of the show's producers and/or personnel at the time) while Takei's and Shatner's were not.
Perhaps the actor believes that taboos were the reason, but actually had quite off-putting nipples that were not seen to be fitting of a Greek deity. (No offence intended to the actor.)
Perhaps Takei and Shatner were given the same tape/makeup treatment, but theirs came out with better definition in the end.
Regarding the third possibility, the following two images (taken from different episodes) suggest that Shatner had nipple treatment at least some of the time. Kirk's nipples are more natural and ample in the first, while in the second they are almost non-existent:
If his nipples were also treated in the first still, then the results of the process were inconsistent, either intentionally or unintentionally depending on the personnel involved and the costuming decisions surrounding those episodes.