Having just sailed Navigator of the Seas after two NCL cruises on Epic and Escape this post is interesting and I really must get around to writing my own review, but for now I thought I'd add comment related to some of the differences noted...
Pre-cruise I totally agree with how complicated I found RCI's cruise planner and "sales" when they change the prices of everything.
Embarkation and disembarkation days were both a breeze on Navigator v both my NCL experiences.
Navigator's pool deck was just gorgeous and I loved hanging out at the Lime & Coconut Bar... quite different to either of the NCL ships.
The food on Navigator in the MDR and Windjammer was terrible. I also really missed having the option of O'Sheehan's later in the evening.
I agree that the RCI drinks package is a little better... but be aware that Starbucks coffee is not included in all RCI ships. On NCL I have had speciality coffee included as I've sailed in the solo studios so that wasn't really a huge advantage for me personally, but having sparkling water included was a nice plus for me.
I found the nightlife disappointing on Navigator and was in bed earlier than either of my other cruises most nights... possibly not a bad thing! haha
Last but not least, I found the crew on Navigator to be the friendliest of the three ships I've sailed on so found it interesting to read the OPs perspective about the Anthem. My cabin stewards on my two NCL sailings were rarely seen and I didn't speak with them... on Navigator I had really pleasant interactions with my cabin steward most days. I also found that bar staff were calling me by my name regularly which was a nice touch.
I've returned to perusing the NCL boards here on Cruise Critic because I think I'll be returning to NCL for my next cruise.