I recently stayed two nights at the St Regis Bora Bora followed by three nights at the Conrad Bora Bora Nui. I've written reviews in the appropriate forums, but thought that I would try to compare the two directly in this post. Hopefully it will help people identify what is important to them and inform any choices.
I clearly had a much better stay at the Conrad Bora Bora, but I will try not to let that sway my comparison. There are advantages to each Resort, but if I had to generalise I would say the St Regis might suit older couples looking for privacy, whereas the Conrad might be better for younger people who don't demand as much space (no easy answer for young Honeymooners, I'm afraid).
Some thoughts on the different elements of the resorts:

• Rooms: The base St Regis rooms are significantly better than the base level Conrad rooms. At the St Regis you get a large villa with a private pool on the beach, enclosed from your neighbours with walls. At the Conrad you just get a suite in an apartment block. In general, the St Regis rooms are larger and more private. In general, the Conrad rooms are (to my taste) more elegantly designed and more modern (also less risk of being run-down which is a problem with some rooms at the St Regis). Like for like the St Regis is generally about 20-30% more expensive, but even so I don't think there is a clear winner here. I had two particularly bad rooms at the St Regis vs. one lovely one at the Conrad
• Clientele: I'm always wary of making generalisations on customers based on one visit only, but when I was there the St Regis was a mix (exclusively) of Americans and Chinese. The average age was older than usual in Bora Bora and there were some families with teenage kids. The Conrad was much more diverse, with many Europeans, Americans, Canadians Brazilians, Chinese and Japanese that I spotted. The average age was quite a bit younger (20+yrs)
• Location: The St Regis is in the top bit of Bora Bora with the best view of Mount Otemanu and the signature Bora Bora water shade. The Conrad is also beautiful and in some ways I think is more attractive (particularly given the elevated area on the island), but the St Regis will win out here for the prestige/ Instagram factor. In real life, I would call this a tie
• Service: The St Regis Butlers are excellent, and provided they don't disappear on you, they will provide a level of personal service that the Conrad cannot offer. The Conrad has some youthful high energy Butler-like people who are fantastic, but also some slightly grumpy staff. The St Regis should win out here, but as I was pretty much ignored for a couple of days the Conrad won on my stay
• Food: The Conrad has a fresher, healthier and more luxurious menu overall, particularly for those who don't like American diets of hamburgers, pizza, etc. The St Regis has Lagoon which is an excellent American fine-dining restaurant as well as the more extensive breakfast buffet. They also have the best pastry chef. Conrad cocktails were poor, most St Regis venues not much better, but at Lagoon they were very well put together. Overall I would call this another score draw, though for me personally as I'd only eat in Lagoon once per stay but often in the casual venues & room service, the Conrad was a better fit
• Environmental awareness and enjoyment: Even for French Polynesia (the French are not particularly careful masters of their territory), I thought the St Regis was a bit careless. They advertised shark feeding which I think was both for captive sharks as well as with an external partner. The lobby TV advertised an underwater personal sub/ mask type thing which showed people treading all over reef as well as guides breaking off coral and even putting sponges on people's heads. There are almost no fish around the St Regis, unsurprisingly. The Conrad meanwhile is a real leader in marine biology and is highly environmentally sensitive including a high profile Biorock product which has resulted in cute fish all around the little reefs around the Bungalows (quite beautiful)
• Status recognition: Rather unexpectedly I got almost no benefit from my Marriott Ambassador status, but a giant upgrade at the Hilton based on my Diamond status. I do not think this is necessarily typical, but I would nonetheless put the St Regis Bora Bora in the category of hotel where you're better off using a travel agent to liaise with the hotel than to rely on your Ambassador