The attraction to the Druid class largely depends on what you enjoy doing. While it's true that a Shaman can heal, DPS and whatnot without changing forms, as RareBeast pointed out, it's not really effective. I have a Restoration/Elemental Shaman and I still need to switch talent builds whenever I want to heal/DPS. The main attraction for me was that the Druid class gave me a combination of the classes in the game - in one class. In Cat form I can pretend I'm a Rogue (with an energy bar and stealth), in Bear form I feel like a Warrior (having a rage bar and tanking capabilities), Moonkin form allows me to toss spells at stuff like any other caster class out there, and Tree of Life makes healing people a walk in the park.
The animal forms gives you so many tricks, it's almost sad.
Cat form - stealth, reduced damage from falling, jumping distances to reach people fast.
Bear form - loads of armor and health, massive damage mitigation and the ability to charge something.
Aquatic form - increased movement speed in water (even better with the glyph) and unlimited breath under water.
Travel form - increased movement speed on land (sort of useless now that mounting takes less time, but Travel form is faster than running and can be cast instantly in combat).
Flight form - increased movement speed in the air (can't be cast in combat, though).
You can't fly in Azeroth(*) yet, only Northrend and Outland, but where you
can fly, the world literally becomes your playground. The day the class peaked for me, I was running away from a silly Horde person trying to kill me while I was questing in Outland. I switched to Travel form to outrun him, noticed he had jumped on his mount (it's faster than Travel form), jumped into a river and swam faster than he could to the end of the river where a large waterfall was. As I jumped off the waterfall, I quickly switched to Flight form and soared away, leaving behind an angry Horde person.
The world (of Warcraft) is your playground.
(*) With the next expansion they're redoing Azeroth, presumably enabling flying there as well.
Edited, Mar 15th 2010 10:09am by Mazra