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.

This is my guess...

First of all, have you tried putting your mouth up close to your skin and testing the feel of both types of blowing? I've found that when we blow on something, we know that we can hit it from farther way if we tighten our lips, like a nozzle on a hose, so we move our head farther aways. But since we can't do that when our mouth is open we move closer. Try your blowing on the back of your hand when your mouth is open, and then without moving, tighten your lips and do it again. In my experience its the same temperature, but its taking up less space of your skin, so you can't feel it as much. This may give the illusion that the air is cooler.

Normally when you blow on your skin from farther away, the air from your body collects the ambient air from around you as it travels to its destination. As it travels it cools down to the room temperature. It may not seem like a long distance to travel for a change in temperature that seems pretty extreme, but even the water from your hot shower is much warmer when it come out than when it hits the floor of your shower.

As already said before your body has a pocket of warm air so you may not notice how cold the room temperature actually is. Imagine that you decided to sprint from one side of your home to the other. The air rushing past your skin would feel cooler because that pocket of warm air is no longer there.