I'd guess that we can feel sunlight because it contains ultraviolet radiation...
We feel sunlight because it's electromagnetic radiation (EMR), and whenever EMR of any wavelength is absorbed by some material, it's energy is converted to heat in the material. In other words, sunlight warms your skin and your clothing, and all of its wavelengths contribute to that warmth.
...which would make artificial light (i.e. light from lamps, bulbs and such) "invisible" to the skin.
Sunlight is bright! Normal indoor lighting is several orders of magnitude less bright (less powerful) than sunlight. At noon in the tropics, Every square meter of the Earth's surface receives about 1000 Watts of sunlight. Compare that to the roughly 100 30 Watts of LED lighting that I use to light up all of the walls, ceiling, and floor (tens of square meters) at night time in the office where I am typing this right now. And, that's 100 W of electricity: Only a fraction of that power actually gets converted into light.
Edit: Revised my estimate of the electrical power used by LED lights in my office.