[Part 1 can be found here.]
After the 16-bit era came the fifth generation of consoles, which gave us the Sega Saturn, the Nintendo 64, and Sony’s PlayStation One. (It also gave us the 3DO and Jaguar, but those consoles aren’t notable for their sound design, on the whole, unless you count the bad acting on FMV 3DO games as part of that.)
The sound cards on these systems, due to memory limitations combined with reliance on formats like MIDI and MP3, had a significant effect on what music could be used in games, a major shift from previous generations that were extremely limited by their hardware. MIDI came into dominance, due to the amount of memory it required as compared to raw pre-recorded audio.
MIDI is a… divisive thing, to say the least. A lot of people are fans of its tones, while others despise it, especially in the wake of the masterpieces of the 16-bit era. It’s a difficult format to work with, easy to program but difficult to get any sort of lush sound. It’s a bit of a stopgap between old technically limited tones from the 8/16-bit eras and fully sampled, pre-recorded music, which makes it hard to enjoy, for many people.
But with this new variance in tones came, as previously mentioned, new variety in what music and sounds could be used in games. Though they sounded nothing like the real thing, many soundtracks went for the film score route, making use of MIDI-based synths to produce works in that vein. Due to MIDI’s divisive nature, a lot of these soundtracks are viewed as missed opportunities, good music with poor sound quality. Personally, I tend to dislike a lot of MIDI-based music, but that doesn’t stop me from appreciating works like, say, the Final Fantasy VII soundtrack or the Metal Gear Solid soundtrack. Whether those works feel too complex for their limited hardware or feel right at home there is up to you.
[Edit: As stated in the comments below, I messed up here. It’s worth noting that the low quality of these soundtracks isn’t due to MIDI itself (it’s just a protocol for communication), but to the instruments used in the production of these soundtracks.]
On the other hand, composers like Koji Kondo, I feel, did an excellent job producing works that fit their limited canvases. Unlike FFVII or MGS, they worked on games with less serious tones, and MIDI tends to lend itself to games like that. Where would games like Super Mario 64 be without marvelously upbeat, fun soundtracks to keep them moving? The same goes for The Legend of Zelda: Ocarina of Time, which also proved that, with skill, one could make MIDI work for more serious games as well.
There were definitely games that didn’t go the “film score” route, however, one of the most notable of which would be the original Silent Hill. Composed by Akira Yamaoka (who would prove to be one of the more eclectic composers to work in the industry), it dealt in trip-hop, industrial, dark ambient, and rock, several genres that one couldn’t really reproduce on previous generations’ hardware. His score formed an atmosphere that was near-impossible to create on previous systems: one of horror, fear, paranoia. If it wasn’t for the advances in sound card technology and the tones possible due to them, the entire horror game genre might never have existed. Sound does much to influence fear, and without the atmosphere created by that sound, fear is almost non-existent.
The sound chip’s ability to use pre-recorded sound samples did much to help this; how else would we have crashes and bangs to shock us in Silent Hill or gunshots and iconic alert tones in Metal Gear Solid? Sound effects became much more important in this generation, since one could easily use sound samples that sounded like the real thing (or came from the real thing) rather than attempting to reproduce the sound with the limited features of that sound card. Unfortunately, most videogames lacked the budget to fully sample the original tones, but this development was a landmark for sound design in the medium.
[Edit: You should read the comments by Rhythmatic and Nick Perry below; thanks for the clarification.]