Final Fantasy VIII felt like a huge step forward in the animation of the Final Fantasy series. The FMVs are still stunning twenty-five years later, but the real difference is in the in-game scenes.
Final Fantasy VIII isn't Final Fantasy's first step into 3D animation; that was Final Fantasy VII, which came out two years earlier. However, Final Fantasy VIII is the first time the story plays out with character models that actually have human proportions, which means the animation can be a lot more naturalistic and subtle.
The graphics in the above screenshot may not be impressive by today's standards, but it successfully conveys both major details - Squall and Rinoa are embracing - and more subtle ones: the two of them have drawn back enough to look at each other, and Rinoa is standing on her tiptoes. When the others arrive a moment later, Squall hugs Rinoa harder before letting go. Even 'two characters are embracing' would have been difficult to get across in any of the earlier games, and none of these subtle details would have been possible.
With the earlier 2D games and the blocky models of Final Fantasy VII, characters usually have to make large gestures for them to come across clearly. When Barret of Final Fantasy VII makes very dramatic arm motions, that's not just because he's a passionate character; it's also because smaller gestures would be unclear on his character model.
In Final Fantasy VIII, they're still not at the point of having visible facial expressions (except in FMVs), but characters can make much smaller motions that are still clear to the player. Silent interactions like 'Zell wipes his hand on his trousers, then holds it out for a handshake; Squall, instead of accepting the handshake, folds his arms and turns his head to the side' would be much harder to convey in any of the earlier games. If a character in Final Fantasy VIII is surprised, they can show it just by jerking back a little, whereas the sprites of the 2D games would have to go cartoonishly wide-eyed.
That's not a criticism of the surprised 2D sprites, of course! They're cute, and they get the job done. With the small scale of the characters in the 2D games, anything else would be difficult to understand. But it's interesting to see the new modes of expression that are open to Final Fantasy VIII's characters.
Of course, later games would have animation that puts Final Fantasy VIII to shame. But Final Fantasy VIII's animation was unlike any of the Final Fantasy titles that came before it, and I think it adds a lot to the game's storytelling.