I have spent many nights sleeping in tents, tarps, and under the stars.
Tents are definitely your warmest option, and can be up to 10 degrees warmer than the outside - if you're using a rainfly. Just using the mesh tent body will even increase the temperature by a few degrees, but won't protect from any weather except mild wind.
Tarps can certainly increase the temperature too, but as @Wills mentioned there are some other variables to take into account. If you attach your tarp low to the ground (reducing air volume and also cross-breeze) you can trap some warmth around you. Planning your placement so that your tarp does not create a wind tunnel, and instead catches breezes broadside, will increase your ability to trap heat. Further reducing air flow by surrounding natural elements or using your pack/equipment to block the doorway works too. In many cases, the disadvantage with temperature is far outweighed by the decrease in weight offered by the tarp. edit: compared to a tent, not to sleeping under the stars.
The shape of the land affects sleeping temperature too - ex: cold air will settle in dips in the ground. Water (streams and lakes) also have a chilling effect on air temperature.
Don't forget that most heat loss while camping actually occurs through the ground (given the calm night you mention). A quality sleeping pad will go a long way in keeping you warm.