Breath is released from your body at 37.5 C 

The temperature outside your skin is normally lesser than that. So, the skin ought to feel warmer when you breathe on it. So why does it feel colder when you "blow" instead of "breathing"?

I thought a lot, and I can come up with two hypothesis.

1) There may be some perspiration/sweat/moisture on your skin, which evapourates when you blow on it. This makes it colder. This is similar to the effect of a hot wind. Even though the wind is hot, if it evapourates sweat efficiently, the net of the two effects may cool the skin. 

2) The second reason is linked with human evolution. Wind has almost always been dangerous for cave-men. If you are out in the cold wind, there is a considerably increased chance of you dying out of cold. 

The body recognises this, and we have evolved mechanisms to recognise pressure on our skin as wind. The brain exaggerates the cooling affect of air on the body if it applies considerable pressure on the skin, and tells us that this can be dangerous.

3) There may be another reason. There is always a warm insulating layer of air near our skin. The wind removes this layer, hence exposing the skin to the colder air outside. Even if the air we blow is warm, after it passes the body, it brings in the cool air from outside. 

Student in Electrical Engineering, Indian Institute of Technology, Kharagpur

I would guess that this was simply due to the wind chill factor.  When you aspirate, you are breathing out a large volume at a low speed.  This air will be warmer than your skin because your internal temperature is higher.  However it is not moving fast, and it is over a larger area of skin, so the energy per square centimetre of skin will be much lower, and it will not speed up the cooling of the skin by evaporation (it may also contain more moisture than would a column of air, and this moisture will be quite a bit hotter than the skin), so the nett result is that the skin feels hotter.  When you blow, the air moves much faster and hits a smaller area, so the energy levels are well over ten times greater.  You can confirm this by holding a strip of paper in front of your mouth and first aspirating, then blowing.  A sneeze travels at anywhere between 80 kmh and 800 kmh, so my guess is that when you blow, it is at least 40 kmh.  This means that air is moving across your skin carrying moisture away and thus making the skin feel cool - the wind chill factor.  At 4 C, a 40kmh wind would make your skin feel like 29F approximately.  This also means that the cooling is reduced when your skin is further from your mouth, owing to the loss of speed.   The wind chill effect is lost when the ambient temperature exceeds certain levels.  For example, where I live it is currently 41C and blowing on my skin makes it feel hot.