I also agree that beginning (and end for that matter) of the season should be avoided. I went on the third week of the season and my boat and some other liveaboards had serious issues the first 2 weeks. Granted, the wet season in the Philippines this year went very very late.. there were still typhoons and lots of rain well into April (not typical) - I was in Western Australia back in October and their "wet" had arrived 5 weeks early. Global warning etc. But I do think you run a better chance of avoiding the worst of the weather by going in mid-April-early May. My liveaboard had to turn back from their crossing in week 1 (they eventually made it but a day late) and we heard reports of another liveaboard leaving Tubbataha early because the weather was getting a bit rough and the boat was careening quite violently from side to side while they were moored. Important to keep in mind that Tubbataha has very limited mooring balls and no land to protect it from incoming weather/winds. Add up the liveaboards and private boats, one side of the atolls being rougher, and you can end up competing for balls or having to leave! (we didn't have this problem fortunately but it was quite windy and choppy at times)
I am not sure that moon cycles make much of a difference in Tubbataha. The park appears to be well protected and cared for - the reefs are thriving and I suspect there is a lot of marine life that is permanently camped out there throughout the moon cycle! From April 8-13 (last quarter of the moon cycle), we saw the "big guys" on all but 2 dives at all dive sites (I was very fond of the east side of the atolls, notably Shark Airport, Amos Rock and Delsan Wreck (this is where the hammers hang out). A good dive master will help - ours knew when and where to go (weather dependent of course) and he was spot on (for example, he told us the hammers would be around 36 m at Delsan and after 5 minutes 6 of them showed up to check us out, knew where to find rays etc). We saw whale sharks every single day at all different spots (morning, afternoon and dusk dives!), several mantas, eagle rays, lots of turtles and literally hundreds of reefies. The other dive group on our boat wasn't as lucky (they were less experienced so weren't hitting the depths like us and thus missed the hammers) and I know someone that was on the Stella Maris the same week as I was and saw whale sharks but not as often as us (and no mantas or hammers). Nobody saw any tigers that week (which I was fine with!) I want to say it involves a lot of luck but a tuned-in DM (knows how to read the currents, where to drop in, experienced enough to know where he usually sees certain species) can also be a huge asset.