Are racials important in Vanilla WoW?

I can't stress enough how I've never ever ever ever played retail WoW, ever, so I have next to NO idea about the game in it's prime (or the game in general). Most games make racials an after thought and they play little to no part in how well your character does but I have a feeling in Vanilla it's a lot different. I've seen a lot of people get really specific when requesting classes (ex. "LF Dwarf Priest") and so I'm wondering, in your opinion, just HOW important are racials? I'm leveling both a Human Priest and a Night Elf Priest, and I wonder if I can just willy nilly level either one or should I focus on one more than the other?