The lore differences:
The Alliance paladins came into existence during the Second War, as the Order of the Silver Hand, led by Uther Lightbringer, used the powers of the Holy Light to battle against the Horde. The Holy Light itself isn't a being or a force, but more of an Eastern philosophy -- the self and the universe are connected, you must work to better the universe, you must respect the happiness of others, etc. However, the actual power of the Light comes from beings known as Naaru. The Order of the Silver Hand remained prominent until the attack of the Scourge, when Uther and many paladins were wiped out by the paladin-turned-death-knight Arthas. The remaining paladins fled to Stormwind and Ironforge, where they spread their philsophy among the dwarves. The draenei had known the Holy Light through the Naaru when they were exiled from the planet Argus, so when they crash-landed on Azeroth, they were all set to be paladins.
Blood elves, however, largely gave up the worship of the Light after Quel'Thalas was destroyed and they needed to feed their arcane addiction. When Prince Kael'thas Sunstrider claimed the abandoned Naaru structure of Tempest Keep, he found one lone Naaru that was left behind to guard the structure. He captured the Naaru and had it sent back to Silvermoon City for study. Some of the magisters began to believe that they could turn the powers of the Light inherent in the creature to serve their people. Lady Liadrin, a discontented blood elf who had quit being a priestess because she believed the Light had abandoned the elves, volunteered to bend the Naaru's powers to her will. She succeeded and became the first Blood Knight. Many other members of blood elf society joined her, believing it their just revenge for the destruction of their city. All future Blood Knights would draw their powers from the captured Naaru.
So there's a huge philosophical difference between paladins and Blood Knights. The paladins revere and honor the Light, even if they sometimes take it too far (see: the Scarlet Crusade.) The Blood Knights, on the other hand, hate the Light for abandoning them and enjoy twisting it to serve them. Roleplaying Alliance paladins hate the idea of Blood Knights, and consider them entirely evil. They wonder how players can philosophically justify the idea of being a Blood Knight. (Personally, as a Blood Knight myself, I consider the Naaru to be stuck-up bastards who are probably just using us in some grand plot of their own. Never trust beings who claim they're here to help you with no strings attached.)
The playstyle and attitude:
There is no hard and fast data about what different races are specced. However, looking at the forums, there does appear to be a certain divide in attitude and role between old-school paladins (human, dwarf) and post-BC paladins (blood elves, draenei to a lesser extent.)
People who rolled a paladin as their first character likely considered it to be the classic ideal of the paladin: A warrior that heals. This playstyle is supported by levels 1-59, where most paladins spec ret to grind quicker and swing around their hammers. Then suddenly, people hit 60, started doing instances, and they were expected to shut up, stand in the back, and heal. While a lot of paladins enjoyed healing, some paladins hated being reduced to healbots, and wanted to have an option to both deal damage and support the party through buffs and debuffs. This group became disillusioned as, patch after patch, the Retribution tree failed to become truly viable in raids. Meanwhile, the Horde grew jealous of what they saw as the paladin's superior buff system and mana efficiency when compared to shamans.
Then along came the Burning Crusade, and belfadins. Suddenly, the Horde was full of paladins -- and most of them didn't roll pallies to hit things with hammers. Rather, they had seen the efficiency of paladin healing and buffing, and wanted to support their guild or raid group through that. Instead of the classic ideal of the paladin, they rolled a pally to do what they had seen the Alliance do for so long. And these healadins often hated the idea of Ret. Why would anyone roll one of the best healing classes in the game to do mediocre DPS?
Paladinsucks referred to these as "Neopaladins: WoW players who rolled a Paladin long after original launch and in full knowledge of their primary healer/support role and endgame caster playstyle. Usually antagonistic towards the few remaining original Paladin rollers who are unhappy with the direction the class has taken. Most recently bolstered by the influx of Horde healing classes rerolling a class formerly unavailable to their faction." It's not a Horde-specific thing, since there are lots of ret and prot belfadins, but more a result of certain paladins who were rolled after BC -- which do tend to be blood elves.
So the old-school pallies, many of whom wanted to be Ret or at least have Ret be a viable option for others, were faced with a horde of elves telling them to lrn2heal. This led to a certain amount of anger towards the blood elves. "You rolled the trendiest, johnny-come-lately, flavor-of-the-month version of a Paladin, and you claim to be an authority on what being a Paladin is all about? Post on an Alliance Paladin, or stfu. Seriously. No offense to other BE Paladins, but posting on one to say anything about what the Paladin class is about is like some dip%*!% who bought his first Red Sox hat after they won the World Series talking about what it means to be a Red Sox fan," wrote Oxyn of Feathermoon. In response, Remuscabalt wrote, "I'm sorry that Blood Elves payed attention to the reality of the game in terms of itemization and talents instead of reading the minds of blizzard developers and getting hard-ons from big yellow numbers." That's pretty much the whole debate summed up in two quotes.
Now, I'm a belfadin. I rolled my paladin to heal and tank after seeing what Alliance could do in raids. Right now, I'm prot, but I'll probably go holy at 70 to help my guild. Nevertheless, I cannot understand why so many belfadins have such a violent antipathy towards improving Retribution. Improving the Ret tree won't create more "retnoobs" that think they're warriors -- rather, improving it will likely lead to more raid utility in the manner of shadow priests. Remember when it used to be "lolshadowpriest" and no one would take them anywhere? Now, because of the improvements, we don't see more bad shadow priests running around -- we see more good shadow priests that can help their group through mana and health return. They're not just sitting around spamming damage spells over and over again -- the people who wanted to do that have rerolled to become bad warlocks, and the paladins who wanted to do nothing but damage have given up and become bad warriors by now. I don't know how an improved Ret would fit in with raids, but I'd love to see how it would work out.
Alliance and Horde paladins, what do you think of your counterparts on the other faction?