The Saturn deserves a mention too, I feel.

It may not be on the same iconic tier as N64 or PS1, but it was still a cool console with a lot of personality and some pretty awesome games; Panzer Dragoon 1 and Zwei, Nights into Dreams, Radiant Silvergun, Exhumed, Burning Rangers, etc.

Definitely the PlayStation. It's home to Final Fantasy 7, 8, and 9, the original Crash and Spyro games, Mega Man Legends, Parasite Eve, Resident Evil, Klonoa, Tomba, Metal Gear Solid, The Legend of Dragoon, Tomb Raider, MediEvil, Digimon World, Dino Crisis, Ape Escape, Parappa the Rapper, Jumping Flash, Twisted Metal... The List goes on and on and on. It had so many great games from nearly every genre. It was the perfect introduction for Sony who was the newcomer back then. That's not to discredit the N64 as it also had some real stand outs. Ocarina of Time, Majora's Mask, Pokemon Stadium, Super Mario 64, Donkey Kong 64, Kirby 64, Banjoo Kazooie, and Smash Bros just to name a few. But in my humble opinion, the selection on the PlayStation is simply more broad and meaningful. I love both consoles and have great memories with both though. :)

PS1 by miles. Ps1 still is my favorite Sony Console and my second favorite non-Nintendo system behind the Sega Genesis. The N64 had its gems but Nintendo was too focused on 3D while the PS1 was literally the second coming of the SNES at the time. I absolutely love the PS1.

N64.

Echoing what people have said, the PS1 had an overall better library of games, but there aren't many I still come back to. Really, Mega Man Legends is the only one, and that's more based on personal nostalgia than anything. N64 only really had a handful of great games (Mario 64, Banjo Kazooie, DK64 sort of, Ocarina of time, Starfox 64) but those are still games I come back to regularly.

It's kind of like two football teams, one of which was generally mediocre, but had a few great seasons and won a few superbowls over a decade, and another team that was a constant contender but never won any. You could make a good argument that either was the better team, but to most fans, those big victories are the ones they remember, fair or not.

PS1 wins this easily for me. It had awesome RPGs and a few awesome 2D games too. On the other hand, 3D action games were a huge step backward for gaming, which made the N64 not nearly as good as the NES or SNES. PS1 gets an easy win in my book.

I almost have to say it's a tie for me between N64 and Playstation. If Nintendo had compromised and agreed to use a disc format with the cartridge format, I think that would've tipped the scales, the fact that it's so close almost entirely on the backs of Nintendo and Rareware mainly is telling. 

PS1. Their 3rd party exclusives like Metal Gear Solid, FF7 and Dino Crisis were untenable. Their first party exclusives like Twisted Metal 1-4 and Syphon Filter weren't half-bad, either.

When in doubt I always refer to my Top 50 list and PS1 wins this one with 5 titles on the list (FFVII, Metal Gear Solid, Castlevania: Symphony of the Night, Resident Evil 2, Tenchu: Stealth Assassins) vs 3 for N64 (Super Mario 64, Mario Kart 64 and Star Fox 64)

PS1 is the best console ever, so it's a real easy one.