1 C atom is just one particle of the unique element Carbon. There is virtually no way to isolate or work with a single atom of anything.
1 mole of C is a quantity, like a dozen or a gross. We use dozens for food sale, like eggs, and a gross for small items like pencils. The power of the quantity a 'mole' comes from the mass of a mole in grams. When you have 6.02 x 10^23 particles, you have the formula mass in grams.
1 dozen eggs will weigh differently than 1 dozen bicycles.
Likewise, i mole of any substance will have a mass equal to the species atomic mass, or formula mass.
So, 1 mole of Carbon will have a mass of about 12.01 g (the number written will depend on the periodic table). 1 Mole of water will have a mass of 18 g (approx)
This is why the mole is such a useful quantity.