What is the difference between ''1 C atom and 1 mole of C atoms?''
What is the difference between one egg and one dozen eggs?  A dozen is simply a certain count, in this case 12, that is rendered as one unit for our convenience.  Avogadro's number s just the same thing as a dozen, except we made the unit count larger than 12 in order to match it up with our conventionally used units.

Your assumptions about a "mole" are thus correct.