Can anyone here enlighten me on why his eyes appear to have a demonic look in oblivion but in Skyrim his eyes appear to be blind.

Whats the story behind it?

I don't know but staring into them is like listening to the Beatles for the first time.

Whomever he is now is not who he was in Oblivion.

There is some speculation that, as sheogorath may very well be your oblivion character, and during Shivering Ilse sheogorath makes a comment about plucking out your character's eyes at the end of the ordeal that this may have actually happened, or your character as sheogorath decided it would be funny to follow up the threat of the daedric prince he mantled.

A Daedra Prince is a concept more than anything else.
They are far too big for mortals to perceive as they really are.
They choose a shape in which to appear to mortal eyes and this is largely how the Daedra wants to be perceived.
Mephala for instance may appear as male or female, depending on what form the mortal (s)he is trying to seduce is more succeptible to.
The form in which they appear is called their avatar, and it is only a small part of the Prince.
Other parts of the Prince are their plane of Oblivion and their artifacts for instance, but they are all just a piece of this greater concept.

Clavicus Vile is an interesting one.
More schizophrenic than other Princes, as his Barbas persona actively works against himself, and so did Umbra.

I think they just made him look different, design choices and all of that.

I don't think, nor will I ever think that he was the hero from Oblivion. If anything once Jyggy was freed from his curse, he was probably promptly re-cursed. Sure he says something about "being there" but that could just be a reference to the Shivering Isles DLC. Or it's entirely possible that they just decided that since he was so popular on Oblivion that they needed to bring him back the way we remembered him, totally disregarding the SI plot.

I could be totally wrong but I doubt the mortal character you played turned into a literal Daedric prince and took on his exact form and personality.

You're wrong

The champion of cyrodiil became Sheogorath, as Arden Sul before him/her did.

As merari explains, daedric princes can choose to show themselves as whatever they please. In the metaphysics of TES, there is no difference between a daedric prince, and a realm of oblivion. The prince IS the realm, this is why a prince can reshape his realm, he and the realm are the same entity - obviously a sentient universe is difficult to converse with, which is why they elect to take more conventional forms to communicate with their followers, but also explains why a daedric prince's form is inconsistent, and why some "female" princes are still addressed in the masculine term, any designation of gender is arbitrary.

this ^^

Interesting, I still don't favor that theory/truth.

I have trouble believing my character would choose to become a crotchety old man with a terrible accent. Or why they would choose to emulate their predecessor identically. Also, if Jyggalag is still Jyggalag, where's the shrine to him? Where's the monolithic crystalline altar in the game, I'm sure in the last few hundred years someone must have started worshiping that oaf.

Maybe the metamorphosis into becoming a Daedric prince so horrible, painful and confusing it's...maddening? That would actually explain a lot...

there aren't shrines to all of the daedric princes in skyrim it could be that none were built.

Jyggalag does not like the mortal plane (too chaotic) and wants nothing to do with it. That is why there are no shrines to Jyggalag.

You'd expect worshippers though. A LOT of people love the idea of order, I find it hard to believe there isn't any wannabe cultists who'd seek to help bring jyggalag's order to the world.

That's what I was thinking, there are a lot of nuts in the world, there's bound to be a few that desire it. Maybe as part of a DLC they might slip a shrine in somewhere, maybe a cave full of beautiful crystals and a giant statue.

You know, I really miss all of the followers that would gather at the shrines.

The problem is Jygg would not acknowledge them. You might have something pretty to look at, but he would not respond.