"We push niches for healers," Ghostcrawler said, "to a small extent, so that players don't just say 'Well, druids are the best healers. Let's take 6 of them.'" It's probably an arguable point, but the specifics of a healer's class abilities are incredibly important, perhaps more than any other raid role.
The healer's job is to make sure that the tanks and DPS don't die. You could argue it doesn't matter which healer class you have, as long as they can accomplish that goal. To some extent that's true. But in addition to simple healing spells, each healing class has very different and specific abilities that make them valuable.
Could you get through Naxx-25 with only a handful of Resto Druids? Probably, but it would be a challenge. For example, my guild has recently discovered how much easier raiding can be with a Discipline Priest along. That's something we didn't have for quite a while, so I'll admit that I'm a little behind the curve here. Bubbles are amusing to watch, and very effective. They are an immense boon to any attempt to kill a boss, especially the ones who hit so hard a tank can't just soak the damage.
Another guild I know has recently recruited a couple very effective Shaman healers. When they're around, they add a very noticeable benefit to the raids. Nothing says "handle the Area effect" like Chain Heal. However, Ghostcrawler goes on to say "We want you to be able to raid if your best holy priest is sick that night by getting a paladin, shaman or druid instead." Blizzard has definitely had to do a very delicate balancing act between making sure that each class is both viable and valuable, and while still making sure that 24 people aren't left hanging if the 25th can't make it to a raid.
I think Blizzard has, for the most part, done a pretty admirable job of that difficult balancing act. Just about every encounter in Wrath (before Ulduar) can be done with just about any combination of healers, as long as you have good players who know their classes.
It's really the general healing abilities which are more important – you need someone with a good AoE heal, someone with a very strong single-target heal, someone with some damage mitigation abilities. And of course, it's always better to have a good variety of classes, so that you get the best mix of cooldowns and abilities.
Except, of course, for Instructor Razuvious on 25-man. That's a special case. (Two priests there, or don't bother.)