Evolutionary advantages of wet nose of cats and dogs?
Why did it evolve ?


  Mammals with rhinaria tend to have more acute olfaction, and the loss of the rhinarium in the haplorrhine primates is related to their decreased reliance on olfaction, being associated with other derived characteristics such as a reduced number of turbinates. 
  
  The rhinarium is very useful to animals with good sense of smell because it acts as a wind direction detector. The cold receptors in the skin respond to the place where evaporation is the highest. Thus the detection of a particular smell is associated with the direction it came from.
  The rhinarium is adapted for different purposes in different mammals according to ecological niche. 
  
  In aquatic mammals, development of lobes next to the nostrils allow them to be closed for diving. In mammals that root, the rhinarium often develops into a hard pad, with the nostrils off to the side. In the elephants it has become a tactile organ. In the walrus, it is covered with stiff bristles to protect it during foraging for shellfish. In many animals the form and purpose of the rhinarium remains to be elucidated.


Essentially it is the case of having a better sense of smell provided by the extended surface area, which has evolved to fit whatever the organism in question needs. I'll update this when I find this in a few more papers.

I took this from the wiki: 
http://en.wikipedia.org/wiki/Rhinarium

How does it work ?


  A model of the canine nasal airway, developed from a 3-D
  reconstruction of high-resolution magnetic resonance imaging (MRI)
  scans, was used to study the aerodynamics of canine olfaction.
  Simulation results reveal that a unique olfactory airflow pattern
  exists within the canine nasal cavity during sniffing that is critical
  for efficient olfaction. 
  
  Calculations show that this novel olfactory airflow pattern provides a
  crucial residence time for odorant absorption in the sensory region
  and promotes spatiotemporal fractionation of odorant mixtures along
  the olfactory epithelium. 
  
  The canine nose functions similar to a chromatograph that imposes
  odorant-specific deposition patterns upon the thin mucus layer
  covering the nasal cavity....The results show that deposition patterns
  are primarily influenced by the intricate olfactory flowfield and the
  odorant solubility in the mucus layer.
  
  Highly-soluble odorants are quickly absorbed near the entrance to the
  olfactory region, and thus do not reach the periphery with significant
  concentrations. In contrast, insoluble odorants are deposited more
  evenly and may even exit the olfactory region without being completely
  absorbed. Predicted odorant deposition patterns correspond with the
  anatomical organization of olfactory receptors known to occur in
  keen-scented (macrosmatic) mammals, providing a mechanism that helps
  explain the excellent olfactory acuity of the dog.


In short the mucus on a dog's nose 'pre-sorts' smells by trapping the passage of some odour particles more than others along the nose, with different types of molecules being picked up at different "depths" in the nasal cavity.

You can read about it here: 

Craven, Brent, Gary Settles, and Eric Paterson. "The Aerodynamics and Transport Phenomena of Canine Olfaction." APS Division of Fluid Dynamics Meeting Abstracts. Vol. 1. 2008.

Lawson, Michael, et al. "Odorant Transport and Deposition in the Canine Nasal Airways." APS Division of Fluid Dynamics Meeting Abstracts. Vol. 1. 2009.

Craven, Brent A. A fundamental study of the anatomy, aerodynamics, and transport phenomena of canine olfaction. ProQuest, 2008.