What is the difference between an atom and a monatomic molecule?How can we identify them?

There's no real difference except in the context in which you discuss them. A molecule is the smallest piece of a sample which still retains the sample's chemical properties. For a glass of water, that smallest piece has 3 atoms (H₂O) so we call it a triatomic molecule; and for a lump of charcoal, that smallest piece has a single atom (C) so we call it a monatomic molecule.

Usually "monatomic" is used in reference to certain gases, in which each each gas molecule consists of a single atom. Helium and Argon are examples of monatomic gases. This is opposed to (for example) "diatomic" gases such as oxygen (O₂), in which each molecule consists of 2 oxygen atoms bound together.

An atom is the basic unit of matter... A monatomic molecule is a molecule consisting of one atom...