Now some would argue that the Horde needs their symbols, and as long as they are peaceful, they should be allowed to use them, But even if you buy that argument (and I don't), the Horde has hardly been peaceful. In fact, they have very clearly been instigators in multiple conflicts. One of the prime hot spots is in Ashenvale.
The main argument for the Warsong Clan's presence there is that the Horde legitimately needs the Lumber. There's two problems with this argument. Firstly, the actions of the Warsong Clan in Ashenvale really can't be justified even if they "need" something there, and secondly, no-one involved even pretends this is about the lumber anymore.
The Warsong Clan first got entangled in Ashenvale by marching into it armed for war, with no idea of who was there or whether anyone else had prior claim to the lumber or the land. Horde apologists will argue that they didn't know it was taken, but this is hardly a legitimate claim. European settlers came to the Americas centuries ago, ostensibly to settle, and for that they needed resources. However, it is very nearly universally agreed that the resulting genocide against the native tribes is one of the darkest chapters of human history. In this case, the natives, the Night Elves, had the ability to fight back with much higher effectiveness than the Native Americans did against the Europeans. This in no way justifies Grom's response. Instead of falling back and waiting for further instructions from Thrall, he attacked the Night Elves with gusto, taking demonic blood to do so.
As soon as the Warsong knew the lumber and land was spoken for, they should have properly withdrawn, or sent emissaries. Since they did not, they very clearly became the unlawful aggressors. To compound this trouble, they no longer even pretend this is about Lumber. Warsong Generals and Tauren Hunters alike call Ashenvale, the ancestral land of the Night Elves, an "untamed wilderness" that rightfully belongs to the Horde invaders, and they now openly leave the lumber they cut out to rot, the very act of cutting down the trees now a sign of dominance and insult to nature and the Night Elves. This is eerily like the "manifest destiny" philosophy the US Army and west-bound settlers used to wipe out or displace the Native tribes they encountered.
In addition, Varian has seen this first hand. After he escaped from slavery, he came to Ashenvale and joined the Night Elves in fighting against the Horde, and saw them summon a massive Fire Elemental to burn the forest. In other words, he was witness to another act of malicious aggression by the supposed peaceful "new" Horde, and had one more reason to suspect that the Orcs were hardly being peaceable, but were clashing with the Alliance at every opportunity.
On the continuing activity of the New Horde: The RAS, the Wrathgate, and the Battle for Undercity
The RAS is one of the other major problem spots in the Horde. At the Wrathgate, their lethality and intent was unleashed, killing many Alliance soldiers, and once again, one of the greatest Generals of the Alliance. This time, it was Bolvar Fordragon, a general as close to Varian as a brother, and acting father figure to Anduin Wrynn in Varian's absence. This is the third time that the Horde has directly and maliciously taken a person close to Varian.
Horde Apologists will argue that the RAS at the Wrathgate were a rebellious splinter faction that should not be lumped in with the rest of the Horde, or even the rest of the Forsaken. Again, there is a problem with that. The RAS was doing exactly what they were founded to do at Wrathgate. Putress' mistake was not destroying the living. It was doing it for the wrong master.
The RAS, from the beginning, has planned to kill humanity. From the first moment that a newly minted Forsaken comes to Brill, he is put to work, not poisoning Scourge, but rather poisoning the living. At one point, Apothecary Johann has you concoct a disease to kill an Alliance citizen, a Dwarf. When he gives you the instructions, he tells you that "this is a subtle hint of what The Dark Lady has planned for the rest of Azeroth." Not "The Scourge." All of Azeroth. And note that he does not say "the RAS," he says "The Dark Lady." From the beginning levels, it is made very clear that Sylvanas is completely on board with the "Death to the Living" part of the RAS' philosophy.
Compounding this is the Forsaken's actions against the legitimate Alliance communities of Southshore and Hillsbrad fields. The army of the Undercity shows themselves dedicated to completely wiping out an innocent farming community in the most bloody way possible, and releases a vengeful Lich on Southshore itself. Mind you, there is every indication that the Forsaken started these engagements. At the very least, Hillsbrad is clearly a peaceful farming community.
Apothecary Lydon makes it very clear that the RAS wants to see everyone dead, and has you use his poisons on innocent Alliance farmers. When you report his findings to Faranell, Faranell is completely delighted in the work and sends you out to gather more reagents for Lydon. It is made abundantly clear that Lydon is not an aberration or part of a splinter group. The entirety of the RAS is as dedicated to killing the living as they are the dead.
If all of this was not enough to incriminate the RAS, the new Arthas novel quite definitely confirms that the RAS is looking to kill the living, and that Sylvanas is in on the deal. A scene set just before the start of the Wrath of the Lich King expansion places Sylvanas in the RAS's chambers, watching as Faranell administers a new plague strain to a young human girl and a stoic Forsaken man (Faranell insists the man is a criminal, but Sylvanas' internal dialogue implies that she doubts he really is but does not care either way). When both die retch and die horribly, she commends Faranell for his work and returns to her chambers, exulting that she may finally be able to kill both the Scourge and the Living with this new strain.
That's right, The Dark Lady is most definitely 100% on board with the killing.
With this in mind, it becomes clear that Putress' crime in the eyes of the Undercity and of Sylvanas was not aiming his Plague Barrels at the living, but doing it at an inopportune moment and for Varimathras rather than Sylvanas. Sylvanas herself is a Banshee, an undead being known for using deceit and charm, and was able to use her refugee caravan to move Thrall's sympathy to her. By all indications, neither Sylvanas nor Thrall has launched an investigation into the RAS' aims and how much Putress got away with and why.
Again, Varian himself has a front row seat to these atrocities. Leading the charge into Undercity, he sees the RAS' inner laboratories, and the horrendous machinery and experiments going on there, experiments that Putress carried out openly, and would have been seen as part of the RAS' stated mission apart from his secret allegiance to Varimathras. Thus, Varian sees that the treachery and maliciousness of the Horde extends not only to the Orcs, but possibly to the other member nations. And either way, he sees the Alliance suffering and dying. Whether this is ultimately because of Horde maliciousness or incompetence doesn't really matter. Either way, it has to be stopped.