To put it plainly:
Yes Amneville has more species, and yes their enclosures may look a little more fresh-faced than the Burgers' enclosures...
But to say that the majority of us are voting for Burgers' just because it's our favourite zoo is completely and utterly unfair. There's a reason (in fact, many) why most consider it to be the best zoo in the Netherlands.
Yes, of course, there is some loyalty to the collection as could be seen by the overwhelming overturn in the Burgers' vs Wroclaw round. But that isn't to say that Burgers' is going to win hands down every time. If it were to face off against a fair few other collections in this competition it would be annihilated.
Plainly and simply, I'm sticking with Burgers' because of multiple reasons. Firstly, as has been said above, they do have some really solid history with the husbandry of primates. First large colony of chimpanzees, a successful breeding history with gorillas etc. etc.
Secondly, the enclosures themselves hold out still today, and that's something that Burgers' is, for the most part, really good at doing; they are constantly aware of what needs replacing and when. The orangutans are gone, and if they were still there in that enclosure then yes, my vote would probably be on Amneville. But what do you have now? You have a visually pleasing enclosure for two species of lemur that also gives them enough mental stimulation and manages to avoid the classic 'walkthrough' that many collections with lemurs use. Then there are the Rimba enclosures, which are well thought out and manage to stimulate the primates just as much, either through exhibit design or species mix. Yes, something should be thought about with regards to the dusky langur/golden-cheeked gibbon mix, but again that could be something that Burgers' is already figuring out how to solve and, at least for now, the two species don't seem to mind rotating. Then, finally, you have those ape enclosures. As CGSwans said, the chimp enclosure is still the best outdoors I have seen to date, and anyone who knows me knows that I've seen a ridiculous number of chimp enclosures. Not to mention Burgers' is one of the few zoos I've been to with gorillas where I have seen such regular interaction between the gorillas themselves and their co-inhabitants (the others that come to mind being Frankfurt, Apenheul and GaiaZOO). And I think that is the key to why I have chosen Burgers'. They have a sort of timeless nature to their enclosures that makes up for the somewhat older enclosures looking a little rough around the edges.
I am not at all saying that Amneville is a weak zoo. I think their primate section is fantastic. But it's going to take more for me than a decent orangutan enclosure, a varied species list and a commitment to keeping gorilla bachelors for me to vote it.
(I am open to being convinced, just like
@CGSwans)