Let John Smith be a normal person, just as smart as you and me. Consider the sentence:
'John Smith will never be able to prove this statement is true.'
It seems easy to prove the sentence true: if JS can prove it, it is false; since nothing false can be proven, JS cannot prove it; hence, it is true.
The problem is that JS, being as smart as any of us, can understand the 'proof' above and then 'prove' the sentence. What has gone wrong?
Most probably, the sentence above is not a real statement but a sentence unable to make a statement or express a proposition; a paradoxical sentence like the Liar, unable to be true or false.
This never happens with formal systems. If S is a correct formal system, it will not prove:
'S doesn't prove this sentence'
beause if S proved it, it would be false and S incorrect, contrary to the assumed.
Therefore, the sentence will be true but unprovable if S is correct.
The difference between JS and S is that, S being a formal system, whether S proves or doesn't a particular sentence is a determinate state of affairs, with respect to which only truth or falsity is possible, but not paradox.
Note that this suggests a difference between provability by JS (or you or me) and provability in some correct formal system.