The games here are
Final Fantasy VII (1997) vs Final Fantasy VIII (1999)
Parasite Eve (1998) vs Parasite Eve 2 (1999)
Resident Evil 2 (1998) vs Resident Evil 3 (1999)
(You could place Parasite Eve between FFVII and FFVIII since it came out between the two and was used to prototype some of the tech for FFVIII)
The leap in quality with the human character models is especially impressive. Look at Sephiroth's beautiful visage here and just think that Square went from that to feeling like their animation was convincing enough to carry an entire sci-fi drama movie with a realistic human cast in less than 5 years
spoiler
Then of course you had Oddworld Inhabitants who came right out the gate with movie quality CGI. Their characters all being fish-faced aliens instead of humans probably helped
To be clear, these are all cgi cutscene assets that never left a graphics workstation. Nintendo 64 is somewhat involved in that Nintendo's insistence to stick with cartridges for the N64 with a max capacity of 64MB made Square and others run away to Sony so they could fill CDs with the beautiful FMV sequences seen here (as well as voice acting, CD quality music, etc)
Absolutely, the N64 was technically more powerful than the PS1, but had less memory for games to work with due to cartrages. Nintendo was actually originally working with Sony to make the N64 a CD system, but after the CDI butchered their characters they chickened out after Sony asked for joint rights to their properties. So basically Sony said "screw you we'll take our CD work we were doing for you and make our own console." At least I think that's what happened, it's been a while since I read about that. But basically Nintendo didn't technically insist on cartrages, they just didn't have the time to make the N64 a CD system after Sony bailed (They tried with an add on in Japan, but it was too late in the N64s life).
Which is the better console is up to personal opinion. N64 had better in-game graphics, consistently good exclusives and had no loading times. PS1 had massive 3rd party support, and CD, which had enough memory for impressive FMV cutscenes and CD quality audio. Also the ability to "please insert the next CD" made long sprawling RPGs popular on the system.
Both had their good and bad points.
It started out way back on the SNES. They were doing joint development with Sony on a CD expansion peripheral (hence that weird port on the bottom of the SNES console) but the project got scrapped. Oddly enough, around the time that Commodore went bankrupt once and for all, Nintendo's name was floated as a possible buyer so they could take over manufacture and sale of the Amiga CD32 system, which was basically just an A1200 crammed into a game console. That just turned out to be a rumor floating amongst Amiga nerds, though, when German business genius tech giant Escom bought out the rights to the Amiga and made the CD32 into the game industry titan that still dominates the global market today.
It's amazing how much business bullshit completely changes the course of a thing while taking none of the public eye