The first question to ask is whether domestic cats have wet noses normally. Under normal circumstances, by which I mean nothing strange climatically, domestic cats do not have wet noses. Their noses are quite dry and I can rely on personal experience - years of it. So a domestic cat's nose is dry and we are referring to the nose leather as cat lovers call it. This is the tip of the nose.
If the nose leather is wet it is probably due to one of two reasons. Firstly: condensation - if a cat is outside in cold weather their warm breath exiting through the nostrils of the nose condenses on the nose leather and when it does so the breath turns into liquid which makes the nose leather wet. Humans encounter this all the time when, for example, going for a walk in winter.
The second reason would be that the cat has a cold or to put it more technically an upper respiratory infection (URI) which causes a nasal discharge, some of which ends is up on the nose leather.
Ah..there is a third possibility. A cat has licked his nose due to displacement activity! That happens quite a lot. When cats are unsure they temporarily remove the uncomfortableness of that feeling by displacing it with a nose lick.
That, as far as I can see, is the full answer. It appears to me that some people think that a nose should be wet because the question is pitched in a way which presupposes that the nose is wet. This is not true in my opinion. It's a misconception. If it is wet a lot for some cat owners then it is because of condensation as mentioned above. It may be because their home is quite cold.
If the home is cold a cat's nose leather will be cold. So when the hot air of the breath meets the cold nose leather the water vapour in the breath condenses into a liquid on that surface. It's just a question of physics. We see condensation all over the place. It happens most normally or likely on windows inside the home or the car.
If there are, say, four people inside a car and it is winter, the breath of these people may condense on the cold windows of the car. It's exactly the same physics. That's enough I think! No, here is some physics.
What is condensation and how does it work? Condensation is the opposite to evaporation, which is when a liquid water becomes a vapour. In condensation water vapour becomes a liquid. When water vapour condenses on a surface it is because the air is cooled to its dew point. The dew point is the temperature at which condensation happens. The dew that we see on the grass in the morning is water that has condensed out of the atmosphere. The dew point varies based on the saturation levels of the water in the atmosphere and the air temperature known as relative humidity.