So obviously the two factions were going to have different experiences hitting the Broken Shore. This is one of the most significant events to happen in World of Warcraft, with very important consequences moving forward. It's also spoileriffic as hell, so let's do a cut.
The Horde actually gets a more complete version of what happens, and ultimately, the red faction accidentally reignites the faction conflict due to poor communication. It's sort of a tragedy, not only because of two very painful losses, but also because it could have been avoided pretty easily.
The Alliance and Horde cooperate in their assault on the Broken Shore - in fact, both sides more or less do the same thing (the Alliance gets to tank Krosus first while the Horde gets adds first.) However, while the Alliance goes off to confront Gul'dan, the Horde goes to flank the Legion up on a ridge above.
What you don't see from the Alliance side is that that ridge has another huge Legion portal, and an even larger swarm of demons that attacks the Horde's position.
As the Alliance fights all the familiar demons we've "killed" in the past, the Horde fails to cover them from above because they're being overrun. Sylvanas didn't betray them - she's desperately fighting for to keep her own forces from being annihilated.
And this is why, sort of like in Romeo and Juliet, one successful bit of communication could hold off the tragic result and the re-eruption of war (though to be fair, it's not clear we're going into full Mists of Pandaria-level total war.) If Sylvanas or her forces had been able to shout down to the Alliance that they were under attack, there would be no sense of betrayal - yes, Varian would likely have still died the same way, but at least there would be no sense that the Horde had responsibility for it.
How dire was it? Well, to hammer things home, Vol'jin, the popular Warchief who saved the Horde from Garrosh's tyranny, is run through with a cursed demonic spear. Sylvanas calls upon her Val'kyr to rescue as many people as she can, but when you arrive back at Orgrimmar, High Overlord Saurfang summons you to an emergency meeting within Grommash Hold.
Vol'jin is dying. The wound is infected with some kind of Fel curse, and he has only a few moments left. The Horde actually gets an additional cutscene where Vol'jin speaks to Sylvanas. No, he's never trusted her, but the spirits are telling him that she's the one who can lead the Horde in this most dire of conflicts. With his last breath, Vol'jin names Sylvanas Windrunner as the new Warchief of the Horde.
And honestly, she seems to take it seriously. Sylvanas is consumed in thought as Baine and Lor'themar carry out Vol'jin's body to prepare for a funeral. As Vol'jin's body is burned on a funeral pyre, she addresses the Horde, asking them for her aid in avenging their Warchief's death.
I've got to say, this is a huge opportunity for character development for Sylvanas. She has done some twisted and evil things in the past, and I don't think her motivations have always been as noble as she claimed. But in the face of Vol'jin's sacrifice, I'd love to see Sylvanas on a road to redemption - a kind of anti-Garrosh plot.
Also, let's not overlook the significance of the fact that the Horde has its first female Warchief. Warcraft has historically been something of a boy's club in terms of major characters, but Sylvanas (despite having a very, very male-gaze-inspired character design - something they've started to work on by giving her more practical leather armor) has been a really interesting and complex character. If they can keep her nuanced - retaining her pragmatic-to-the-point-of-callous methods but also growing her appreciation for how much the Horde has helped her and how responsible she is for both her own Undead faction and the larger group of living Orcs, Trolls, Tauren, Goblins, Blood Elves, and Pandaren, we could see some really great things.
But there's no way that this helps with relations with the Alliance. Even if there had been no accidental sense of betrayal, there's no living leader within the Horde (well, undead, but close enough) more likely to stoke Alliance hatred (especially with Genn taking a more prominent role, finally.) Vol'jin was a fantastic choice for Warchief when it was time to end the conflict between the factions. The troll had a history of working alongside the Alliance, even sending emissaries to Stormwind to seek aid in fighting the Zandalari. If there was anyone the Alliance could trust to keep the peace, it was Vol'jin.
But Sylvanas? Hoo boy. Even if she commits herself fully to avoiding conflict with the Alliance, the mere fact that she has ascended to this level of power will probably provoke the Worgen at the very least to consider themselves back at war with the Horde.