Growing up, some of my most important gaming experiences were turn-based RPGs. Final Fantasy 6, Chrono Trigger, and Earthbound were three of the most important games to me of the Super Nintendo era. These days, the turn-based RPG is in decline. Square-Enix themselves have abandoned the TBRPG format for their flagship Final Fantasy series. They aren't gone, certainly, and there are new turn-based RPG releases, but they are fewer, further between, and not as 'big' as they used to be. Why?
I'm sure there are several reasons, but I think that one of the big ones is the increased strength of modern consoles allowing for better artificial intelligence.
The main appeal of the classic turn-based RPG (TBRPG) was that you had a party, and savvy players would exploit the synergy created by their party members to make a strong group. Furthermore, the draw of the TBRPG was the complex stories and character development that you got by having a large roster of characters.
In order to have the multiple characters all act, this used to require their actions to be selected by the player, since having the computer choose the actions intelligently would have consumed processor cycles that couldn't be spared. Since choosing actions for multiple characters in real time would be difficult for the player, a turn-based system was adopted.
With more powerful consoles, it became more possible for the console to come to intelligent decisions regarding movement and action, not only for the enemies but also for your allies. This freed up the player to just control one character, allowing them to create a greater depth of control and gameplay for this one character.
The turn-based RPG game does stand alone as a great type of game, one that I enjoy immensely. But it's good now that it's no longer required in order to do a party-based story.