Bringing threads like this up from the dead can be very confusing because all three lines have modified their product; Azamara the most, then Oceania and, to a lesser degree, Princess.
The real fascination for me is the one saying that I constantly use in the luxury cruise market: "It is the software, dummy!" In other words, the hardware (ships and facilities) are important, but the differences that really matter tend to be on the service side of things. Great service with lesser facilities (think smaller Seabourn ships) can be more attractive than one with all the bells and whistles, but which lacks service...so there are constant frustrations.
As is mentioned in another thread, Azamara has been a cruise line trying to find a home and a message. It has, by those that even know of it, been perceived as "Oceania-esque", but it is making a real effort to differentiate itself by becoming more inclusive, increasing the levels of service and becoming ultra-port intensive. One interesting marketing concept is Azamara insisting that while it is more inclusive it is not a luxury product. That may relate to the size of its standard cabins, but I think it is more of "underpromise and over produce"...which has been a successful strategy for Oceania over the years.
Oceania is pretty much an ala carte experience (you pay for most everything) and little niceties such as complimentary shuttles into town were excluded. BUT, here we go again, it is now starting to include those shuttles in more ports. It is, well and truly, a solid product with a loyal following...and great itineraries. It also stresses its cuisine which is, from the accounts of my clients (some of which also sail Seabourn), acceptable, but clearly not on the level of a luxury line.
Princess uses the R-Class ships as decidedly Princess products. The service, charges, etc. are pretty much as they are on its bigger ships. The level of service is OK, but not near the level of Oceania or Azamara. But its pricing also reflects that.
So be sure you obtain a good understanding of the products today and not those of even a year ago.