I had both and my games for both were quite different. Definitely loved the N64 for games like Zelda, Goemon, Jetforce Gemini, etc., and I think the only true RPG I had for it was Hybrid Heaven, which I loved for the combat system and not much else. Oh yeah, Goldeneye and Turok were legit amazing on there too, can't forget those. Of the vast PS1 library, the majority of the games I had were more "last generation" or "arcade" in spirit, like shooters like Einhander and Thunderforce or games like SNK's Metal Slug and King of Fighters '99 and that was about it. So I don't have a lot of experience with PS1 games. However, some of this does make sense a bit, because the PS1 just could not keep up with arcade ports for some reason, in snapiness, frame smoothness, and annoying load screens abound. King of Fighters '99 is a great example, it felt nowhere near as smooth and you'd spend just as much time watching load screens as you did playing. But I know a lot of arcade boards, besides being chip based, also had the appropriate amount of RAM to help them out, plus I've been told the PS1 wasn't designed with sprites in mind, so that probably didn't help. While I never owned one, I've heard The Saturn handled sprites much better. It's a shame it wad so hard to program for though.
Edit: Speaking of gameplay, the PS1 controllers felt much more appropriate for platform type games while the N64 controller, weird as it was, was better for 3d games. This was especially noticeable in games like 1080 or Road Rash where a slight difference in tilt made all the difference in the world between success or failure.
Coincidentally, next gen I had a PS2 and a Gamecube, so I was continuing the trend of double dipping and I loved both for different reasons, but the two felt much more similar than previous generations. That said, Gamecube graphics felt cleaner than PS2, but the PS2 had the better controller and a lot of the games felt snappier (and there were A LOT more games overall too).