Fatal Frame was definitely one of my favourite franchises at the time, in fact, just about any of the major or lesser known survival horror games were.
The third game in the series is always going to be one of those overlooked, late generation titles in my eyes: Kuon, Haunting Ground, Rule of Rose, Cold Fear and this game all seemingly went under the radar despite the popularity of the genre at the time. Some may argue 2004's Resident Evil 4 changed expectations, but that doesn't happen in just a year or two.
What I loved about Fatal Frame III was that it tried something "new". I put new in quotation marks because while it might be new for the series, I always felt it borrowed heavily from Silent Hill 4, but in a good way. SH4 experimented with creating a "safe space" to which the player returns to in order to break up tension and also gameplay. While back here, exploration and gaining information took precedent, it was the "adventure" portion of the game if you will.
FFIII did the same thing, even more in fact. While back in the real world, this is where you would gain information about the characters and mansion by exploring the records in your own home. Sending the player back to a familiar, homely setting was genius in my opinion. It succeeded in contrasting the tension / fear of the unknown of the main game with moments of relaxation in familiarity, that is until both games chose to shake things up later in the game by letting the paranormal bleed into the real world.
I'd recommend Fatal Frame III to just about any survival horror fan, it's just that good.