There's not much different worth talking about for whichever you side with. But the main thing is there will be 9 shrines in the Solitude temple if you go with Ulfric. The Imperials are banning worship of Talos, which means there are only EIGHT shrines if they win the war
Eight is pretty much returning to a past Imperial tradition anyway. That's why the Prophet in TES4: Oblivion's Knights of the Nine expansion referred to it as "The Eight and One".
As we all know, the Imperials already worshipped the Eight Divines since the religion was instituted by Alessia so long ago. The Ninth Divine, Talos, was added to appease the Nords whom were conquered by the Empire during Tiber Septim's time.
Does that mean that the eight makes more sense? Lore wise, I mean
As the official Imperial religion? Yes, the Eight Divines makes more sense lore-wise.
The "Nine Divines" change was made after Tiber Septim died in the early Third Era. The religion in its original form has been in Cyrodiil since at least the mid-3rd century of the First Era.
Eight Divines were there since beginning of time in Elder Scrolls. The Eight are the Aedras who sacrificed themselves to complete Nirn, the world of Elder Scrolls.
The human god hero Talos appears later and it is legitimately Ninth Divine. The Thalmor hated the fact that it was a human who achieve godhood before any Altmer. So they wanted to undo Talos by completely eliminating him by depriving him of worship which is technically where the Gods get their power to influence Nirn. (Altmers follows the Auriel's teachings to seek perfection so they eventually become Gods.)
They wanted to deny his existence with a bunch of excuses, for example Ninth Divines is not right or nonsensical, there's only Eight Divines etc... for referrence just go speak to a Thalmor in the game. They will say something similar along with you humans are inferior, Elves are superior therefore you should obey your Elven Masters without question talk.
There used to be 8 Divines. But after Tiber Septim became the Human God-Hero Talos, it became 9 Divines... argued by some that Talos is stronger as he didn't sacrifice a part of him to build Nirn.
I also believe opening talks about which side to join as he has difficulties choosing.
Make no mistakes, the Empire and the Stormcloaks means nothing to me. If the opening asks me, I would say either side doesn't matter.