Why the breath sometimes warm and sometimes cold? (2 different explanations!) If you blow air against your hand with your mouth open, you feel warm breath. If you do with with your lips closed except for a small opening, you feel cold breath.
One explanation from here says "With your mouth open most of the air which reaches your hand has come out of your mouth. With pursed lips lots of cold air is entrained so the air reaching your hand is cold and moving quickly."
Another explanation from Paul Hewitt video,(go to min 24:00), is that air comes out cold it because it expands coming out of small opening.
So which explanation is right?
 A: I think the expansion explains only part of it -- a small part I'd say. 
I think the more likely explanation is that blowing with your mouth open results in an airflow that moves much slower than an airflow blown through a small opening in the lips. The faster moving air will cause a quicker rate of evaporation on your hand, which makes it feel colder. 
But proper measurements are always more convincing than theories :) 
A: It's a simple enough experiment to do.
I took a kitchen thermometer and blew hard on it for as long as I could manage (about 30 seconds). With the thermometer about 15cm from my lips I got half a degree temperature rise. With the thermometer 1cm from my lips I got a 5 degree temperature rise. A quick check blowing on my wrist from 1cm confirmed that at this range the air feels warm.
So either the air current I produce is turbulent and mixes with cold air, or the air current is cooled by the surrounding air as it travels through it. Cooling due to expansion is ruled out unless there is a significant pressure gradient from 1cm to 15cm away from my mouth, and this seems unlikely.
I'm not a smoker so I can't easily examine the air flow to see if it's turbulent. I would guess that it is, but not very turbulent because the air current feels fairly localised even at a distance of 30cm. So I suspect the air current from your breath is cooled by the air around it.
A: Do you have a thermometer?
You could test if the air is really colder or if it just feels colder
A: Probably wrong, but could it be that your breath is pulling a current of ambient air with it, which could explain why when you open your mouth more to exhale it creates a larger warmer current, and can more effectively insulate from the outside current around it. this could be tested with a thermal camera most effectively. but again probably completely wrong. (based on high school level education) 
A: Ruling out
It's not to do with the speed that the air is coming out of your mouth. You can test this by blowing really lightly with your mouth almost closed, and then by blowing as hard as possible with your mouth wide open (so that when your mouth is closed the air flow is less fast than with the mouth open): it's still warm with your mouth wide open and cold with your mouth almost closed.
My theory
I suspect that because of the small gap when your mouth is nearly closed and the curved lips and then also different surrounding air pressures, all these would cause most of the air that's flowing from your mouth to spread out, which means that less molecules (albeit still a lot) are hitting your hand than when blowing with your mouth wide open.
Because there are less molecules hitting your hand at any one time (also less densely) you inevitably have far less energy hitting your hand as well, which in turn means less heat. Because there are more molecules hitting your hand at any one time (also more densely), there is more energy hitting your hand as well, which in turn means more heat.
Summary
So to sum up what I have just suggested, which seems to me a very liable suggestion, the difference in temperature on your hand is to do with the fact that there's a difference in the number of and density of molecules hitting your hand at any one time. 
Note: I may have seemed quite ambiguous when talking about the "density" of the air hitting your hands. When I say "more dense" I mean that there's more $air/mm^2$ hitting your hand.
A: The temperature is the same either way. It is crucial to remember that your skin feels heat transfer, not temperature. There are two things to consider:
Size of opening and amount of hot air: at the same (wind) speed, the current from a large opening will be larger than from a small opening. This means that when you blow breath from a widely open mouth, more hot air is coming out of it to warm your skin compared to the case where you make a little current from slightly open lips.
Speed and evaporative cooling: if you are blowing hair from a widely open mouth and then tighten your lips without reducing the effort, the current of air remains the same and thus the speed of the breath increases a lot. The faster air will cause moisture on your skin to evaporated, thus cooling you and making you fell chilled even though the temperature is the same in both cases (this is how fans work).
A: I will give contra example for some of the comments:
You can blow with mouth wide open very hard and then the speed of air is high but it is still worm. Also you can blow with lips pursed very gently and the air speed is slow, but still you feel it is colder … 
so the right answer is : air comes out cold, because it expands coming out of small opening.
A: Breathe out onto your hand, is it warm or cold? If you're a healthy human, it's warm. Blow on it, and you'll feel it as if it's cold, but cover your mouth with your hand, and blow as hard as you want, and it'll still be warm, from the feeling of it, around about as warm as just breathing. 
Knowing this, it's likely the journey of the air after it leaves your lips that determines it's temperature.
