Zombies, mummies, ghouls, skeletons and other undead are usually depicted as rotten, but vampires are always lifelike. What's the reason?
As Hank mentions in this video, zombie creation myths (like superhero origins) are associated with what one fears in the society those myths are created. We fear that zombies will infect us with disease/modified-genes/something else X and this is shown outwardly by their fetid nature; we fear inheriting their weaknesses, often created from our own hubris.
Likewise classic vampires often represented an analogy of the aristocracy, being kept as apart from others. Consider the classic Dracula from Bram Stoker's novel. His 'disease' is one of wealth and status, sucumbing to it is not a matter of destruction but of separation from other humans. This subset of horror is considered 'the sublime', a concept developed in the 18th-19th century.
Burke was the first philosopher to argue that the sublime and the beautiful are mutually exclusive. The dichotomy is not as simple as Dennis' opposition, but antithetical to the same degree as light and darkness. Beauty may be accentuated by light, but either intense light or darkness (the absence of light) is sublime to the degree that it can obliterate the sight of an object. The imagination is moved to awe and instilled with a degree of horror by what is "dark, uncertain, and confused." While the relationship of the sublime and the beautiful is one of mutual exclusiveness, either one can produce pleasure. The sublime may inspire horror, but one receives pleasure in knowing that the perception is a fiction.
More modern vampires, whilst not 'suffering' from wealth and status are often still loners or those who must stay away from others, lest they take advantage of the less powerful. They are shown to be immensely strong, beautiful or otherwise powerful. If you look hard enough you'll find that (unless I am mistaken) Twilight was first supposed to be a mormon perspective on celibacy; by making this choice Edward separates himself from others (and indeed even from others of his kind, sort of spoiling the metaphor, though the difference between the two are distinct).
Now that's not to say that vampires can't rot, as shown by rcollyer's answer; if the vampires were to rot however, we'd be discussing a different trope, perhaps corruption from power.
In short it's a distinction that writer makes between the horror of those beneath us and the horror of those above us.
This is simply not so. Vampires have often been depicted as corpse-like. Pale cold flesh. Long fingernails and hair, longer than they were in life... as is commonly (but erroneously) believed to happen to corpses in their casket. Barring those stories where they "sparkle in the sunlight", they've been depicted with foul breath, the stench of death, and every other form of corruption save maybe visible decomp. Whatever charms they do have, many authors depict these as some obscene form of mesmerism where their victims imagine (due to wishful thinking?) that they're still alive and ignore all the (to the reader/audience at least) obvious signs that their loved ones are dead and something evil has taken over their cadaver.
In those stories where the lead vampire, a Count-Dracula-like figure, is more publicly presentable, this seems to be mostly the absence of odor and (somewhat) normal behavior. The pale skin is often still apparent to those who bother to look.
You are right that we never see actual decomposition, but then even in zombie movies do we rarely see this. The Dawn of the Dead remake shows victims who are mutilated and still animated, but only in the end credits do we see one that is undeniably rotten and still moving. The Walking Dead television series has shown only 3 that I can think of at this point, and one was only partially rotten (this is important, because in reality the still living can exhibit such necrosis/gangrene). While I do agree that the cliche/trope/concept is of a slowly degrading corpse, that's become a much less popular depiction in recent years as pathogenic explanations are favored above supernatural explanations.
It's debatable even in the traditional views just what the cause is: do vampires decompose at all? Only at a slower (imperceptible) rate? Is this due to their higher level of sophistication (a zombie can't even turn a door knob, but vampires are more than capable of chasing and catching their prey no matter how much they try to evade)? Is the necromancy that brings them to their un-life stronger in the vampire, or does the sophistication alone allow them access to greater magic? Both feed on living flesh, only the favorite tissues differ. It raises all sorts of other interesting questions, like what happens to a zombie after 50 years? Are they animated skeletons? Does the animation slowly give way into something more akin to proper death? We know the answers for vampires (they wait out the centuries), but I'm unaware of a story that answers it for zombies.
As the comments mention, there is a great variety of different answers to this question, depending on the universe.
In some universes (Twilight and Scott Westerfeld's Peeps), vampires are not actually dead, but are humans who have been modified by biological or supernatural processes. Not being dead, these "vampires" would not rot.
In most universes where vampires are undead, they feed off of living being and so do not normally rot.
As Keith mentioned in his comment, one of the vampire groups in the Dresden universe does rot. Also in the Anita Blake universe there is a line of vampires that can rot and then reform themselves.
In the Anita Blake universe, there is a case of a vampire who falls in love with a human and, out of deference to her, limits his feeding to animals. As a consequence of this, he begin to develop a rotting condition that eventually affects his mind and makes it necessary to destroy him.
In some universes, there are "flesh eating zombies" that, by consuming living flesh, are able to maintain themselves. While their condition doesn't typically improve, they also don't rot any further.
The other vampire courts operate somewhat differently, and can at least appear completely human. Although, not much is known about the Jade Court.
If I may weigh in on this one question here. I agree with K-H-W in the above answer that is does depend on the fictional universe. However, according to The Vampire Diaries, if the vampires in this series do not get fresh blood, human or animal - but human is preferred, will begin to desicate - they begin to rot and go dormant until they come into contact somehow with some fresh blood to restore their bodies to their original Vampire form.