[Indranil (OP)]

As I know one-mole of CH4 molecules = 6.023 X 10^23 CH4 molecules and one molecule of CH4 = one C atom and 4 H atoms bonded together.
the molecular mass of CH4 = 16 amu and the molar mass of CH4 = 16g/mole
The what is the difference between one mole of CH4 and one molecule of CH4? or there is no difference but the unit only? Please explain it. Please point out my mistakes if I am wrong.

[Bored chemist]

A mole is like a dozen, or a score.
It's fiddly to buy eggs  one at a time so we box them up in packs of 12.
It's fiddly to work with molecules  (or atoms) one at a time, so we box them up in groups of 6.0221409e+23.

That's a peculiar number, but it has one advantage.
It means a mole of hydrogen atoms weighs one gram (almost exactly) and a mole of carbon atoms weighs 12 grams (again, almost exactly).

Since a mole of methane is made from a mole of carbon and 4 moles of hydrogens, it figures that it weighs 16 grams.

[Ophiolite]

Just to add to Bored Chemists neat explanation, I suspect part of your confusion arose from the similarity of the two words:
mole and molecule

The reason for the similarity is that mole was derived from the German mol, itself an abbreviation for Molekulargewicht, molecular weight, in English.