The same thing happens when drying wet hands using an air dryer. The user thinks "If only warm air came out of this machine I would get my hands dried much quicker". It is only when the hands are nearly dry does the air warm up and you feel the true temperature of the emitted air. This is due to a "change of state" or "Latent heat of evaporation". Water turning to vapour requires a lot of heat and this comes from the air dryer heating element. The same thing happens when you blow hard on your skin. The warm breath evaporates sweat and cools the skin. Slow moving air does not evaporate much sweat and it feels warm.

The 'cold' and 'hot' affect is due to the air pressure. Your breath is naturally warm as it has been in your body which is a comfortable warm temperature. When you breathe slowly out onto your hand then you feel the warmth of your breath.

When you blow fast on to your  hand then you are creating a low pressure system in the stream of air, this low pressure system causes the moisture on your hand to 'boil' at a lower temperature, in the process using heat your hand has to make the transition from liquid to gas. The removal of heat from your hand means your hand gets colder.

The same affect can be observed with a fan. A fan will cool you down because it generates a low pressure system, or stream of air, that helps your sweat evaporate and remove heat from your body.

Wind chill factor means that air flowing faster will feel colder.  Like when you blow on hot soup or dry your hands under a fan, more air exchange means more heat is transferred to the surrounding area so warmth/water are lost quicker.

A high-speed breath is blown out through pursed lips and under pressure.  The stream of air is fast and narrow.  Due to both of these factors, any heat held in the exhaled air is quickly lost to the surrounding air, and the breath feels cold.

A slow breath, for example trying to warm your hands in winter, is exhaled through a more widely open mouth.  A larger volume of air travelling at a slower speed does not lose thermal energy so quickly and therefore feels warmer for longer.

Further, it seems that the volume of air expired actually has a greater effect than the speed of exhalation on the temperature of the breath; an open mouthed, as-fast-as-possible exhalation is still warm, and it is quite difficult to blow slow enough through pursed lips that the breath feels as warm as any wide-mouthed offering...