Garrosh Hellscream is many things. He's brash, headstrong, arrogant, concerned for his people, determined to deliver the whole of Azeroth into their dominion no matter what anyone thinks about it, but one thing is clear. He's not Gul'dan. For all the grief I like to give Horde players (mainly because it's easy to rile Horde players up, I know, I raided as Horde for all of Cataclysm and a good chunk of Mists) It's true that on the surface, the Horde of today has changed greatly from the Horde Gul'dan created.
The Horde as it exists today is the spiritual successor of the Horde that Orgrim Doomhammer created when he seized power. Was Doomhammer a kindly, soft spoken orc who loved kittens and rainbows? No. No he was not. He was an orc who had come to power as the right hand of Blackhand the Destroyer, a hunter and a warrior who had spent his entire life in combat. He was strong, devoted to his people, and absolutely committed to an orc victory no matter the odds. In a way, minus Garrosh's bluster and bravado, the orc he most resembles from the history of the old Horde is Orgrim Doomhammer.
The problem was and still is fairly simple. The Horde as it exists now dates back to Gul'dan's need for a structure to shape and direct the nomadic clans in a singular purpose. He did not invent the Horde, precisely. There were traditions dating back to the distant past when the orcs were fighting for their lives against the gronn and their ogre descendants of a unified force, commanded by a single orc, that could come together and deal with the ancient enemy. When Ner'zhul convinced the various orc clans to make war on the draenei (misled as he was by Kil'Jaeden) he didn't try and organize them any further than simply giving them directives under the auspices of his position as the most renowned shaman of the orcish people, but Gul'dan saw the need for organization both to placate Kil'Jaeden and as a means to more surely ensure the end of the draenei people. It didn't hurt that with the decision to sell his people into demonic slavery for greater personal power Gul'dan realized that the orcs would become impossible for him to directly control. They were already a proud and aggressive race of nomads who disliked the idea of a centralized government and who engaged in conflict both with the ogres and among each other. Even before they drank the blood of Mannoroth, they were unlikely to follow someone like Gul'dan.
Gul'dan's masterstroke was the creation of two organizations. To those former shaman who lost their power to command the elements he offered the Shadow Council, a means to the demonic powers of the warlock. To the vast majority of the orcish people, whose souls now burned with the unholy frenzy of the blood curse, he offered the Horde. Led by Blackhand the Destroyer, a physically powerful orc with ambition that far exceeded his intellect, the Horde would serve Gul'dan's plans without the danger of actually being seen to be in charge. Meanwhile, the Shadow Council of warlocks would give Gul'dan a means to direct the occult might of the Horde as he saw fit, and served to allow him to manipulate the Horde from behind many guises. And for the entirety of the war against the draenei as well as the invasion of Azeroth, Gul'dan was satisfied with the arrangement.
It wasn't until he got greedy that things fell apart. The Shadow Council created in Garona an agent who would execute their will, infiltrate the humans (as a half-draenei, she was conditioned to believe she was in fact half-human) and ultimately strike down the human king, Llane Wrynn. It was all going according to plan until Gul'dan's silent partner Medivh was struck down by his former friends and allies Khadgar and Anduin Lothar, and Gul'dan attempted to telepathically steal the secrets of Sargeras from the dying Guardian's mind. Instead, he nearly died with him. In so doing, Gul'dan exposed the true weakness of both the Shadow Council and the Horde.
The Shadow Council could not direct itself absent from Gul'dan. He had ensured that his was the driving will. There was no consensus, no loyalty between council members - each served at Gul'dan's pleasure and intrigued against each other, which kept them from uniting and attempting to overthrow Gul'dan himself. Similarly, with the Shadow Council at each other's throats, Warchief Blackhand was incapable of directing the Horde minus their input. He wasn't a strategist, he was a jumped up thug better at crushing heads than using his own, more cunning than smart. It was this moment of chaos, when Blackhand couldn't find his footing, that Doomhammer exploited. He challenged his Warchief and tribal leader to combat, killed him, and assumed control both over the Blackrock clan and the Horde entire. While a united Shadow Council under Gul'dan could and would have taken steps to remove Doomhammer or bring him in line (the very reason he hadn't moved against Blackhand before) in their fragmented state, with Gul'dan lying comatose, Doomhammer easily removed their subtle control of the Horde by killing most of them.
Ironically, in the very moment of their destruction, the Shadow Council's agent did indeed murder King Llane Wrynn, paving the way for the orcish Horde to destroy Stormwind. Still, Doomhammer was now Warchief of a Horde committed to conquest on an alien planet, one that had under Gul'dan and the Shadow Council left their own world tainted and dying. Doomhammer never even considered negotiation with the humans. It's possible that his world view was simply too shaped by Draenor, a hostile world where the only path to peace with the ogres had been their subjugation, or that the war against the draenei had shaped him without his realizing it. Despite never having drunk demon blood, despite his personal misgivings (whatever they may have been) Doomhammer committed himself and his people to the destruction of the humans who had never offered him or his people any insult, just as he participated in the annihilation of the draenei. And he did so using the same methods as the Horde had under Gul'dan. Indeed, he did so using Gul'dan himself.
This is why I said before that on the surface the Horde today has changed greatly from Gul'dan's Horde, because at the very moment that Doomhammer killed Blackhand and assumed total power, he brought Gul'dan into the ranks of the Horde and made use of his own slain Shadow Council as the first of his new death knights. The position of Warchief, a figurehead when Blackhand held it, now held real power but Doomhammer never used that power to reform the Horde in any substantive way. He didn't seem a return to shamanism, but instead put warlocks like Gul'dan in positions of power. Gul'dan was even allowed to lead an entire clan, the Stormreavers, in addition to his loyal servants among the Twilight's Hammer. (It's telling that, with Gul'dan dead, only the Old Gods could command the loyalty of the Twilight's.) And yes, in the end Gul'dan betrayed Doomhammer and the Horde, as everyone always knew he would.
It was this failure on Doomhammer's part to purge the Horde that resonates to the present day. Thrall's attempt to preserve Doomhammer's legacy has allowed Gul'dan's original Horde to exist like a cancer eating away at the Horde entire - even Garrosh, who emulates Doomhammer in his ways, preserves this 'Victory at any cost' mentality that allows Gul'dan's legacy to continue. As long as the Horde will do anything for victory, Gul'dan will remain as a presence within the modern Horde.
While you don't need to have played the previous Warcraft games to enjoy World of Warcraft, a little history goes a long way toward making the game a lot more fun. Dig into even more of the lore and history behind the World of Warcraft in WoW Insider's Guide to Warcraft Lore.