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
Yeah, they started using FMVs in 1999 because MPEG-4 came out and they could fit them on a CD.
I believe before that there weren't that much high quality 3d FMVs. But there were some high quality 2d animation and stuff with real actor, but those were mostly for PC games where they could divide a game in multiple cds.
Sony was getting devs set up for PS2 as well. They were really hyping the emotion engine stuff and probably wanted their main developers to be able to hit the ground running when it came out.
The FMVs made at the end of the PS1 era were supposed to be a teaser for what the PS2 could do real time.