Great question! First, let's start by clearing something up: the white fog you see rising from hot water or flowing off of something really cold is not actually water vapor. Water vapor is an invisible gas (the air is full of it!), so you can't see it. That white fog actually consists of tiny water droplets floating in the air (like clouds).

With that cleared up, let's get to your question: the reason both hot and cold objects emit that fog is actually the same: when you cool down air, it can't contain as much water vapor, so the water condenses into droplets (which make that fog). So, when room-temperature air is cooled down by touching a cold object, some of the water vapor in it condenses and forms white mist; that's why you see white mist flowing down from around cold object. Similarly, when air touches a hot liquid containing water, it heats up and absorbs some of the water in the form of water vapor. But as it moves away from the hot liquid (by rising up from a cup of coffee, say), it cools back down to room temperature. As it does that, it can no longer hold the water vapor it absorbed, so the vapor condenses into water droplets and forms that white mist again.

Vapor are just molecules (or atoms) of the object that have broken free of the chemical bonding (whatever form of it) that holds them into the object. Because individual molecules all have energy, they all can break free, but the rate at which they do so depends on the probability that they will have enough energy to break free, and this depends on the average energy available to them, i.e. the temperature. So cold objects do not evaporate nearly as quickly as hotter objects, but all objects evaporate to some extent unless of course the rate of accretion of new molecules from the surrounding medium exceeds the rate of evaporation.