What difference would you expect if the “and you know it” clause were omitted?

Suppose there are indeed some people who would clap ”if they’re happy”, but refrain “if they’re happy and they know it”. The implication would be that these people are happy and don’t know it—and yet we’re supposing that they would have clapped “if they’re happy” anyway.

If they didn’t clap, adding the clause would have no effect and its inclusion would have no motivation.