I've had this discussion on here before - there are, of course, spectacular private villas. Both in hard product and in service. Some, such as Iniala (villa resort rather than single private villa, but point stands) seem to go above and beyond anything others can offer (I haven't stayed there, just from reports from this forum).

On the other hand, it can also go so sour, so quickly (see DSI's report of Ani Villas in Anguilla, a brand which was previously praised to high heaven - something which is very and most unlikely to happen with somewhere like Aman. Regardless of thoughts on old vs new onwership and whether the brand has lost its 'soul', Aman has managed to maintain a fairly good level of consistency in delivering its product and service).

That's before you even get to the general inconsistency of villas - because it is so entirely dependent on owner+villa manager, you really need to pick these out very carefully with a lot of research - which, for me, goes way beyond looking at TripAdvisor etc.

I just stayed at a 4-bedroom villa (20,000+ sq ft, 20m private pool) in Phuket. In description, I could easily write that it was much the same as your villa in Bali - it had a private cinema, a sizeable back garden, private BBQ and the works. In reality, it was absolutely wretched, one of the worst places I've ever been to and stayed at, and it was entirely down to the villa manager. However, I have no faith in the brand (of villas) as a whole because you're only as strong as your weakest link. As a result, I have no real intention of ever going back to any of their properties. Bear in mind this was a complimentary stay - I dread to imagine how they treat their paid guests! On the other hand, something like Treville in Phuket I can imagine being highly rated.

Even on the point of Phuket, even Phuket has some absolutely stunning private villas. I'd much rather go to Amanpuri personally.

Of course there are other things to consider - how many bedrooms do you need, budget etc. Villas can work out great for larger families and groups (Aman has their own villas too, at several times the price - so in the end it's a question of budget).

That's before you get to the overall discussion of villa vs resort/hotel - where you have more communal and general facilities, more dining options etc. In the end though, that's a preferential thing. Some generally speaking prefer one over the other.

But Bali Amans, Bali Amans are a whole different category. I think some of the others have already expressed why - I'll start by saying that it's the decor, the design. It's absolutely timeless. Then it's the locations (bear in mind these are old resorts - when not much was around in Bali. Some still retain their charm - Amankila, others such as Amandari and Amanusa have seen much grow around them). Ideally, rather than considering them as singular resorts, especially on a first trip - I'd highly recommend doing all three on a circuit. But what really makes it, and I will summarise it, it's the staff. Many have been at the respective resort since its inception, going back more than two decades.