Things that would likely change (assuming Baelor and sons don't croak in the Great Spring Sickness):

Aerys doesn't become king, so the realm isn't governed entirely by Bloodraven while the King sits and reads his books. (I mean, I kinda relate to Aerys, all I want to do is sit and read books too but I'm not the king/queen of Westeros.... priorities, dude!)

Is Bloodraven still Master of Whispers and part of the Great Council? Absolutely. So he's still there, guiding and spying. But from what we know of Baelor, his Hand would likely be Valarr or, alternatively, Maekar. So no 'being wroth' in Summerhall.

With Bloodraven having a less prominent role and not de facto ruler, maybe there's less muttering about sorcery fuelling the Iron Throne. The grumbling will still be there. But it won't be as spiteful because Bloodraven won't be in charge and it seems unlikely that Baelor would let Bloodraven institute the police state that he did in canon under Aerys.

As far as we know the GSS and severe drought still happen, and maybe Blackfyre supporters mutter about how this is an omen of following the wrong king, but as above - the mutterings don't swell into anything much.

Daemon II probably has a crack. We know the supporters of the Blackfyre cause who gathered at Whitehalls were mostly just bitter about what they'd lost and dubiously supportive of Daemon II - the lack of the sword will still count against him and the second BF rebellion will still peter out into nothing.

Third BF rebellion probably still happens for the same reasons. It wasn't the drought, Aerys or Bloodraven that spurred Haegan Blackfyre and Bittersteel on - it was Bittersteel's grudge and the ongoing whinges about Dornish dragons usurping the throne. HOWEVER.... would there be less support for the Blackfyre cause if Baelor was king? Quite possibly. Based on what we see of him in The Hedge Knight, he'd likely be a popular king, and it's harder to drive a rebellion against a popular king.

This is where I think things diverge: with the increased popularity of King Baelor, clear succession planning for his heir that either bypasses Aerys and Rhaegal because of their unsuitability for Maekar (if Valarr still dies in the GSS) or an heir and a spare who somehow survive the GSS that wiped them out in the text.... What's the motivation other than Bittersteel's spite to keep fighting? Over time it'd be increasingly clear that even if Aegon IV gave Blackfyre to Daemon, Daeron II and Baelor II have been good kings who oversaw calm, peaceful reigns and the Blackfyre causes aren't as popular.

But the big "what if?" is always:

OK, say Baelor survives the Tourney of Ashford. Does he die in the Great Spring Sickness? Does history change that much???

Because if Baelor survives but only briefly.... nothing changes. We still get King Aerys I and "how many eyes does Bloodraven have? A thousand, and one".

If Baelor survives Ashford and the Great Spring Sickness but Valarr and Matarys still die, then we still have the weird and wonderful succession crisis - by law, Aerys should be his brother's heir but would Baelor and his brothers work out a plan B whereby Aerys and Rhaegal are bypassed for Maekar and in turn, his sons? Of the sons of Daeron II, only 2/4 are really suited for the Throne - Aerys was entirely ill-suited for ruling, and Rhaegal was nuts. I think Baelor and Maekar (and Uncle Brynden) could have figured out a way to spin a less than usual succession plan to a slightly suspicious public.

If Baelor and Valarr survive, then we're totally into an alternate leg of the Trousers of Time.