Cities change things, that's just the way it is. Old ways fall away and new ways rise up, usually while the past and the old ways of doing things lose their original meaning and become idolised without a proper context.
Urban Fantasy is more or less based on this idea, along with the obvious 'things are not what they seem' angle that's common to the genre. Monsters, which in horror would have single-minded and rather shallow motives (kill humans, possess humans, feed off humans etc, etc) suddenly are part of communities and have to deal with issues like politics, or love... Or paying rent on a space to live, Other issues become prevalent as well, can a werewolf really risk hunting and killing prey inside a city's borders, can vampires rely on nightclubs for their blood when there's a shortage of virgins to go around? They change and new priorities arise as they interact with other communities and with protagonists (who may or may not be monsters themselves, but we assume given the genre are aware of the superhuman crew and the places they hang out in the city's 'backstage'. Monsters can play many roles, as a result. In Urban Fantasy's close cousin, Paranormal Romance, the Byronic and moody/angsty elements are emphasised though never to the extent that a plucky heroine can't step in and save the day through the power of true love. In many Urban Fantasy novels, the politics and intrigues of monstrous communities come to the fore, with vampires plotting against each other or wizards acting as a sort of arcane police force. This brings crime into UF's purview, which is a natural fit not only because cities and crime go hand in hand (sheer population numbers dictate that) but also because if you have vampires, werewolves and so on engaging in their own private affairs, there's bound to be some overspill into the human world. If there isn't, arguably we don't have a story.
The other thing we must realise is that, as a result of the tonal shift from horror to something else, whatever that is, the nature of monsters has to change. A vampire who could destroy a party of protagonists in a horror story will be scaled back to give UF protagonists a chance. In the same way, werewolves are cut back in strength and have more complexity added to their lives. We're not so interested in the horrible things they can do, but instead in how they interact, the stories of treachery and so on that almost inevitably arise out of monster politics.
And yes, politics. I don't mean the bland or idiotic stuff that dominates the mortal world but the sexy intrigue type of politics where vampires jostle for position in court and faeries stab each other in the back. It's integral to Urban Fantasy, especially in a more kitchen sink setting, the sort of thing that Dresden Files or The Hollows books provide. In worlds where Masquerades and Veils have been abandoned in favour of monsters living out in the open, such as Mike Carey's Felix Castor books, you have the added issue of how human authorities deal with the presence of inhuman monsters, which can mean anything from Ministries of Magic to sanctioned hunting teams that go around killing monsters for fun and profit. The question of how mortal governments deal with monsters is one fraught with problems though, and most authors shy away from the issue in part because nobody wants to read books where there are a lot of bureaucratic meetings and not a lot of adventure (well, unless you like the Laundry Files books which handle the horror of administration work very well indeed). My point is that the monsters become more human and take on more human concerns as a result of living in cities and having to deal with members of their own species, as well as members of other monstrous factions. This is one thing that UF does extremely well and which reflects the nature of minorities in the real world, giving the genre a foot in reality. In terms of adding flesh to the bones of what are often one-dimensional beings, it's pretty obvious that adding a civic element is a good way to go.
Adding to that, the nature of the city allows monsters to have normal lives (or as normal as they can be when you turn into a nine-foot tall war beast every full moon), adding more depth to characters. As a result, they can become well-rounded and fascinating to read about. This isn't to say that the characters in horror are blank, many are well conceived and written, but there is a limitation to what you can do with them when all they are is the opposition. As most contemporary fantasy works present more slippery 'shades of grey' worlds that have less well-defined boundaries, it stands to reason that your more human monsters have more scope.
Probably the most popular monster of our times, the vampire finds itself adapted to a series of new roles, from seductive madames to stealthy assassins. The traditional role of nobles lords lends itself well to CEOs and the idle rich, whose wealth derives from suspiciously obscure sources of old money, of course. Vampires are perhaps the most easily adapted monster to our modern world, we can picture them strolling through the city at night, drawing crowds, finding willing victims who long for the sensation of fangs in their neck. Failing that, modern resources allow vampires to get blood from blood banks (if they have the right sort of clearance) and to do little harm. Louis, in the Vampire Chronicles series, need not simply feast on rats, he can get a pint of type o from the local blood bank. Everything that we picture humans do, we can picture vampires doing, just more easily. They are the 1% of Monster World.
In contrast, werewolves seem to be doomed to blue collar work, mechanics and the like. The importance of pack is often emphasised in the genre, in a direct contrast to the solitary loup garou you often find in films. This is not a lonely curse anymore, but one where you're almost haunted by the fact that there are others like you, often seeking to establish who's stronger in the most violent fashion. The problem is no longer that you can't trust yourself around other people because you might flip out and hurt them, but that an unwillingness to fight may be construed as weakness. This can be a double-edged sword, on the one hand, a pack is useful, it gives you people to rely on, comrades and allies. On the other, well, it can start to feel rather cheap as a reader when everywhere you look there are more lycanthropes and the curse isn't just limited to wolves (how many kinds of changer do we need, one for every creature?) Call me cynical but when you have werebuffalo or whatever, I feel it's a bridge too far.
Of course, in contrast, Faeries are so diverse that you can pretty much throw a shoe and hit whatever you want, whether that be shape changers, blood drinkers, or something else.The nature of the Fae is basically so diverse that you could pretty much build a ten part series using nothing but them and not run into repetition, or risk boredom from overuse. Perhaps, hypocritically these are my favourite monsters, I think they're beautiful and their nature is one that seems so at odds with humanity, not in the form of predator and prey but in the sense of the Fae almost living a completely different life to humans.
I feel that we're using monsters as metaphors even more, it's just that vampires are no longer just a metaphor for bestial sexual hunger, but also for the ease that the rich pass through life. Werewolves are a metaphor for the rage that men can feel, and for territorial nature of gangs. And it's pretty cool, that we can use these creatures to talk about the issues we have.