It's much more than just their immediate appearance, although that in itself can be quite nice sometimes. I personally particularly enjoy 'badass' looking dragons, rather than pretty ones. Anyways, dragons, like several other fantasy races, are awesome because they represent limitless possibilities. You have the species, of course, but then you have to take into factor things such as longevity (how old the dragon is), cunning (smart dragons are much cooler than dumb feral ones) and power. It's all in how the dragons are handled, some can shapeshift and live basically forever, affecting the world around them and sometimes even manipulating it. While others... Just kind of sit on a pile of gold in a cave until they get murdered by adventurers.
That's the gist of it, to me, it's all about the possibilities, though dragons are hardly one my favorite fantasy races personally. It's all about them being a race that can pretty damn imposing and impressive. Though I do agree that the garden variety feral fire breathing lizards are rather dull and serve as little more than cannon fodder, albeit dangerous cannon fodder.
Dragons can be an extremely destructive force of evil, or paragons of 'goodness', and can shift the events of the world around them dramatically.
Probably my favorite example of this is the Breath of Fire series, which showcases shapeshifting, intelligent dragons in various forms. 3 and 4 in particular do it really well, especially 4 as it showcases the difference between a dragon on the side of good (The main character, Ryu), and one on the side of evil (The ancient emperor, Fou-Lu) even going into their motivations and background.
Then you have the Warcraft universe, with dragons that are literally are so ancient they represent various aspects of the world they inhabit, such as time, life, and so forth, and where one such dragon nearly destroys the world. Granted, the way WoW handles Deathwing is kind of lame but everything prior to that was really cool, particularly in the novels.
So yeah. In short, it's all about the why, how, and whatnot. Dragons can be super lame, or super cool. But this goes for basically most fantasy species with long lifespans, like elves, demons, and so forth. Dragons are just another example.
Oh and undead dragons of various types are super cool looking. Dragon zombies, skeleton, ghost, etc. Heh!