Most Horde characters are fighting Alliance NPCs before they get out of their starting zone. Orcs and Trolls fight soldiers at Tiragarde Keep. Tauren kill Dwarven miners in Bael'dun. Blood elves kill night elves and a dwarf. The Forsaken don't encounter the Alliance until Hillsbrad, but hoo boy do they make up for lost time.
Alliance PCs start a lot later. Draenai have blood elves in their starter zone, but those work for Kael'Thas, not Silvermoon. So they and the night elves don't run into actual Horde until they get to the Warsong lumberjacks in Ashenvale.
Humans can make it all the way to Outland without dealing with Horde, assuming they avoid the battlegrounds. Elwynn and Westfall are crowded with human rebels. Redridge has non-Horde orcs. Duskwood has a couple of Forsaken but the questgivers have never heard of them. The Stranglethorn trolls are also in non-Horde tribes. Heck, the Hinterlands has a PvP quest chain for the Horde but the Alliance ignores the Horde village. Finish up with Searing Gorge, Burning Steppes, Felwood, and Winterspring and you're ready for Outland without a questgiver ever asking you to kill a member of the Horde.
Dwarves and gnomes have a similar situation. The Dun Morogh trolls aren't part of the horde. Loch Modan has some wandering Hordies but you're not asked to go after them. The Wetlands orcs are again hostile to everyone. Same for the Tauren in Dustwallow Marsh.
This makes a certain amount of sense given the human opening cinematic: Stormwind's armies are fighting the Horde and all other problems are being delegated to the freelance adventures. Presumably Thrall doesn't have the cash to support so many regulars and has to bring in mercenaries to get the job done. But it gives the world a secure feel for Alliance characters. Humans know there's a war off in the distance, but the big worries are traitors and the random nasties Azeroth is always producing. Your home is safe, the war is being won by the professionals.
For Horde characters the war with the Alliance is right in their backyard. There's enemy troops near your home and there aren't enough troops to go around so PCs have to fill in the gaps against the enemy. That's a different experience in a more dangerous world.
Do other players get this feeling? Does the world feel different when you're leveling Horde or Alliance? I have to wonder if Blizzard did this deliberately or if it's just a coincidence.