In my youth, waiting for the school bus in a Wisconsin Winter, all the local kids would “see their breath”–see the condensation, caused by moisture & low temperatures.
Since smoking was not stigmatized back then, we often tried, with all our might, to use our breath to “blow smoke rings” using our quite visible breath (but no tobacco).
None of us ever did. 
Why not? 
Wow.
Three hours +, & not so much as a nibble nor a snark.
Have I stumped the SDMB?
Oh, Mighty Cecil Adams!
Will you answer?
I’m going to guess that it has something to do with the particulates in smoke, but I ain’t no scientifical type.
Or maybe you just weren’t holding your mouth right.
Here is the WikiHow. Basically, you take the smoke into your throat and push it out in small amounts in a particular manner. Thus it is the smoke itself that is being manipulated. This is not the case with your breath’s condensation in cold air.
It can be done, but it’s a matter of technique, not of “all your might”.
I’m a Californian, so my cold weather experience is limited, but generally, we can only expel visible “fog” by expelling warm, moisture-laden air directly from the lungs, with the mouth and throat open and the tongue held out of the way. Manipulating the air flow with the mouth and tongue, as we do when blowing smoke rings, gives that air time to equilibrate with the environment in the mouth. The mouth is somewhat warm and somewhat moist, but not enough to produce “fog.” I suppose you can do it if it’s cold enough, but I’d guess that you still need to minimize the time the air spends in your mouth.
Mere lack of particulates is unlikely to be the reason. You don’t need smoke to make vortex rings. Smoke/fog just makes them visible.
We need someone in a colder clime or with access to a walk-in freezer to test this. Take a simple home-made vortex ring generator into the cold air and see what happens.
You can definitely make a vapor ring with cold air. I think your technique was off.
I think the difference is that the smoke and your breath can be separated.
But if all you’ve got is breath, everything that comes out of your mouth will have the same humidity and temperature, and can’t be easily separated.
Supposing it’s possible, it wouldn’t last very long. Once your breath chills, it’s invisible. Your breath typically disappears in a second or two in frigid weather.