As the candle burns, its wax does more than simply melt. The liquid wax rises up the wick and vaporizes in the tremendous heat of the flame. It is this vaporized, gaseous wax that mixes with air and burns. The wick itself contributes very little to the flame; you are instead looking at burning wax vapor.
When you blow out a candle, the top of the wick remains extremely hot. You can see its incandescent glow for seconds after the flame is gone. It is still hot enough to vaporize wax and also to continue reacting with oxygen in the air via a flameless burning process—the blackened wick is effectively a piece of charcoal. But without an actual flame, most of the vaporized wax doesn’t burn properly. Instead, it condenses in the colder air around the wick to form first a mist of tiny liquid wax droplets and then solid particles.
The smoke that you see is primarily these unburned wax particles. They are tiny, translucent spheres that float around in the air like dust particles. They have a strong smell that most of us find irritating, so we like it when the candle’s wick stops burning quickly and cools off so that the wax vaporization process ceases.