storage space for CD quality audio, storage space for full-motion video, on-board full motion video decoders, cheaper media costs, easier media to deal with on development side, and most importantly a game library that catered more towards 'adultish' fans.
I remember reading back when these consoles were current that the biggest draw for most consumers was the fact that full-motion-video was a first class citizen on the PS1, and it was a huge technical wow for people. FMV was difficult on the N64 due to size and codec constraints, so it was rarely done -- and when it was if was usually subpar compared to the PS1 version of the same game. Resident Evil franchise was notorious for this phenomenon.
There was also a supply issue early on in the n64's lifecycle, I seem to recall it being just shy of Tickle-Me-Elmo hard to find.
The ps1 was also easier to pirate, double edged sword though it be.
Mod chips.