What is the difference between a radical and a neutral alone atom?
A chlorine radical and a chlorine atom are effectively the same thing. This does not mean that "radical" and "atom" are interchangeable as jargons. They are used in fairly particular contexts, namely


Saying an "atom" is a reference to an instance of a particular element. They are often discussed abstractly, in terms of their constituent parts (nuclei and electrons) without coming to discuss their  surroundings or conditions within a system.
It is natural to say molecules are made of atoms, because they conserve the total number of nuclei and electrons with respect to the individual atoms, so we can rationalize them as bonded atoms. But saying molecules are made of radicals is weird.
Radicals can (very often) be polyatomic, like a methyl radical $\ce{CH3.}$
A radical has an odd number of electrons at its outer shell, it is chemically unstable, although electrically neutral. Atoms can be neutral and chemically stable, as in the noble gases, which fully comply to the octet rule.


Thus an atom being a radical and vice-verse is more of a "chemical coincidence" than anything else, kind of like a hydrogen cation is a proton. But they are definitely not equivalent.