Reading other comments, a few people have mentioned that the Sony name was huge in electronics back then. This is true but, in the world of gaming, Sony was just a mid tier third party.

As I said before, a lot of big companies tried to jump into gaming but gamers were mostly loyal to the two companies that (appeared to have) dealt exclusively with gaming. Nintendo and Sega. Sega was busy committing suicide by not supporting the Genesis or its expensive add ons.

Early Sony games like Wild Arms and Twisted Metal did okay (EGM even gave it game of the year over Chrono Trigger) but the PS1 was basically sold on its 3rd party library. The biggest games that raised eyebrows were Ridge Racer, Battle Arena Toshinden, Wipeout, Tekken, Mortal Kombat 3, Street Fighter Alpha, etc. and there was no reason to keep any of them exclusive.

PS1 won because it had tons and tons of games that the N64 didn't, it had the best versions of games that the two shared, it had cheaper games (It was also super defective).

Launching late really hurt Nintendo but launching with a more expensive format that guaranteed worse games just handed the industry to Sony and they've run with it ever since. By 1998, it wasn't even a battle anymore.

People here think that Nintendo would have been the dominant market leader in that era simply because of a format change. Not entirely true. The truth is, Sony as a big influential company had many companies flocking to their side from Sega and Nintendo, simply for the fact that they were A: new and had a fresh and amazing marketed console B: Incredibly trusted brand C: Tokyo based, instead of Kyoto based. Sony did their utmost to lovebomb the hell out of these studios. Things would probably still go the way it did. As Japanese devs switched, so did western devs. The format is just another reason beneath this big shift of Sony entering the market, but not the main reason for them become dominant.

Not entirely true. In fact, Square was working with Sony before there was any idea of FFVII. Secret of Mana was supposed to come out for the SNES-CD addon, but due to reason known, the game had to be cut significantly to go on a regular SNES cartridge. There are sources that even say that this very experience is one of the if not the main reason why Square went along with Sony, after they broke off and made their own competing console. Square continued with Nintendo's SNES for a while because.. well.. it was the best selling console in Japan by far. 

This was not just a format change that made Square switch, but a more business and cultural one. Both of them got fairly close as companies and Square didn't release any game (other than FF) for the N64. 

FF VII made PS1 relevant and other games that depended on the CD to have the cutscenes may have not gone to PS1. That wouldn't be a reality I would like to live on.
But Nintendo relationship with several companies were already on shake basis so perhaps not all would change.

! I don't think I've heard that particular story before. Interesting bit of history. I'm now reading an article about the SNES-CD. 

I don't understand why so many votes for N64 victory. There would be definitely a great battle in America, where Nintendo could win in the end, if we take under consideration the base from SNES and NES years. But, in Europe...no way. The very best 64 could have done, would have been around 10 to 15 million, and again this is very optimistic. In Japan with FF and DQ, obviously would have won. There would be no reason to buy a PS1. But then again it would lose in Rest of the world. So, the most logical opinion I 've read is the 80-50 with PS1 victory.

I've heard a smilar story but about Final Fantasy VI, when Square intended a 32MB cartridge for the game but Nintendo only gave them 24MB, so Square had to cut content on every corner. That's why some side characters seem to be out of place like that pickpocketing wolf in Narshe or the swordsman Siegfried, also Uematsu was forced to cut some original songs entirely, some graphics and designs had to be scrapped and some story events were rewritten. Sakaguchi and the top management at Square were very unhappy because apparently this decision came very late in the development process and so this ultimately led to deep disagreement between the two companies which once used to be close partners.

So almost everything I will say has already been said but let's be clear it is confirmed the FF7 started development on the N64 but was jumped to Sony when the storage medium couldn't do what they wanted. CDs would also have made it much easier to get 3rd parties on to the system and with the systems stronger hardware could have lead to massive games on discs. Lets also factor in the 64 disk drive from Japan its possible many of those games would have just been released normally which could have given Americans game like Mother 3, animal crossing and F-zero X with a level editor. Would this have been enough to fend off Sony's head start? Maybe.

I suspect it could have changed everything drastically. Had Final Fantasy, Resident Evil, Metal Gear, etc. all been multiple platform... with the N64 being more powerful. The cartridge move was a bad one for Nintendo. It stifled development and increased costs. I remember paying $70 for N64 games and $40 for ps1 games.

Microsoft is bigger and more powerful than any of them, but they barely squeaked into 2nd place with the Xbox, came in 2nd with the 360 only after Kinect, and is going to finish in third place this generation to a system that came out two and a half years after the Xbox One.