This is a tough one, so I'm going to break it down.
Gameplay
Both games have their strengths and weaknesses. Let's take a look at the two main areas both series have in common, combat and puzzle-solving.
In combat, Zelda wins, hands down. There is just something more satisfying about actually swinging a sword at just the right time while dodging an enemy's attacks, and the options to switch things up by using a lot of different pieces of equipment, like an ice beam or a hookshot, some of which are actually required to defeat certain enemies (albeit usually just bosses).
There's a reason Ys eventually started copying this idea to an extent, and that's because combat in the early Ys games, with the exception of certain bosses, was really quite boring once the novelty of the combat system wore off. There's no variety. It's almost completely stat based. Once you discover that you can attack off-center without consequence, that's about it as far as strategy is concerned, except for bosses. In most cases, if you can't beat the enemies, you just go back to the previous area and grind until you can. Most enemies are a joke if you've leveled properly, because once you hit them, they sort of get stun-locked and you can just blitz them until they die. The Fire Spell doesn't help, because again, there's no strategy with normal enemies, just fire until dead. Now, if I recall correctly, there's a couple enemies that are fire-resistant, but they are VERY rare. You also usually have so much MP that it becomes easy to exploit fire magic on the majority of enemies that can be killed with it. Combat in the early Ys games is almost always braindead simple.
Zelda also wins as far as puzzles go. The puzzles in Zelda are intriguing, hard to figure out at first, but once you know the secret, it usually seems obvious in retrospect. And while a lot of the puzzles fall into the trap of 'use most recently obtained item here', in most cases, new surprises are rampant, keeping the player on their toes.
The puzzles in Ys, by comparison, are generally simple roadblocks opened with key items. In most cases, what you have to do is more obvious than in Zelda. Most items are only used once or twice in Ys, whereas most items in Zelda usually have many uses throughout the rest of the game. There are a few REALLY clever ideas in Ys, like turning into a monster and speaking to them, which are so brilliant and well-executed that it makes you want to cheer, but those moments are few in quantity.
In general, Zelda wins in the gameplay department. Ys has its clever moments and unique ways of doing things, but Zelda's gameplay is much more refined and intelligent.
Presentation
Both Link to the Past and Ys I & II have graphics that are serviceable, but usually not feasts for the eyes. Ys edges Zelda out due to the advantages of CD hardware, though. The cutscenes in Ys are great feasts for the eyes, and every dungeon looks different. Zelda has...text monologues, and most of the dungeons, if not all of them, use the same tileset with a different palette, and perhaps a few unique elements here and there. Later Zelda games would fix these problems, but here, the limits of the SNES, and cartridge games as a whole, really show.
Audio is a very similar story. Both games have impressive soundtracks, but Ys wins due its CD advantages. Almost every area has a unique music track, and there is not one bad track among them. Zelda has excellent music as well, but less of it. There is a song for the light world dungeons, and one for the dark world dungeons. Both are...OK. Neither ranks among the better songs in the game, which is a problem because the dungeons are where you'll be spending most of your time. Ys also has the 'wow factor' of competent voice acting. Zelda only has text.
Story
In this category, Ys blows Zelda out of the water. In fact, it blows the modern Zelda games out of the water! Every Zelda game has followed the exact same template. You always know what to generally expect. And this template wasn't that interesting to begin with. It basically boils down to saving the land by collecting mystic artifacts.
Ys I falls into the mystic artifact trap too, with the collecting of the Books of Ys, but it still had a dynamic plot compared to Zelda. You were always learning more about what was going on. It set up mysteries early on that were eventually solved. Where did all the silver go? Who is the man in the black cape? Is he the mastermind behind the monsters? Just what DID happen to Ys? It is this mystery that keeps the plot of Ys alive, whereas Zelda's plot is secondary to the action. Ys also has an emotion to the plot that Zelda is lacking. The characters in Ys are drivers of the plot and its impact, whereas Zelda's characters rarely move beyond stereotypes and givers of side-quests. The plot of Ys is also just plain original even to this day, whereas in Zelda's plot every game feels like a retread.
Replay Value
I have only played Link to the Past all the way through once. I have played Ys I & II multiple times, in both the Turbo version and the Eternal/Complete/Chronicles version. While later Zelda games were much better at adding things to keep you playing even after completing the game proper, LttP doesn't have much to say once you've beaten it. There are a few easter eggs and hidden items, but the game itself is such a long, hard haul that I've never really desired to go back to it in earnest. Ys has that special something, that perfect combination of elements that can't be explained, that really makes you want to experience it over and over, even though you're just playing through the same game again. It doesn't make logical sense, but Ys wins here.
Overall Impression
Let's face it, while LttP was a great game for its time, it hasn't aged as well as Ys in many ways. The story is trite, cliched, and formulaic. The dungeons are monotonous in their graphics and sound. In retrospect, it's just another Zelda game. It's not bad, but later games in the series have surpassed it to the point where it's not worth playing today.
The reason Ys stands the test of time so much better is something that I cannot fully explain, but it has to do with being more than the sum of its parts, and almost every part is well crafted. Even though LttP has "better" gameplay, it feels like much more of a slog to play through today than Ys. There's something about Ys that just makes it stand apart even today.
Ys wins.
