What is difference between “true in ZFC” and “true”? Let $P$ be a claim. There are two questions.


  Is it true in ZFC that $P$? 


And the other is: 


  Is it true that $P$?


What is difference between these two questions?
I suspect that "true in ZFC" is intended to mean "provable in ZFC."  The relationship between that and just plain "true" is more a philosophical question than a mathematical one.  Let me explain two of the many possible philosophical positions, in the hope that they give an idea of the range of possibilities.

(1), a version of formalism: In mathematics, we have no way of knowing things except by proving them.  We accept, more or less arbitrarily, ZFC as the standard basis for our proofs.  So when something is proved in ZFC, we are willing to say that it is true.  It makes no sense to speak of truth apart from provability.  Therefore, "true" simply means provable in ZFC.

(2), a version of Platonism: There is a world of abstract entities, including the sets and related entities studied in mathematics.  To say that a mathematical statement is true is to say that it corresponds to the actual situation in this ideal world.  We have only a partial understanding and intuition of the ideal world; much of that understanding is summarized in the ZFC axioms, so anything provable from those axioms is known to be true.  But there are other truths that we may not be able to prove and, indeed, may not have any other way of knowing.  Therefore, "true" is a considerably broader notion than provability in ZFC.

To see how these viewpoints work out in a nontrivial situation, consider the fact (provable in ZFC) that, if ZFC is consistent, then it neither proves nor refultes the continuum hypothesis (CH).  A formalist (or, more precisely, someone who subscribes to the version of formalism that I've described) would conclude that, if ZFC is consistent, then CH is neither true nor false.  (He might reconcile this conclusion with the law "p or not p" of classical logic by declaring CH to be meaningless.) And he might have a problem with the issue of consistency of ZFC, because, if ZFC is consistent then this fact is not provable in ZFC.

A Platonist would begin by brushing aside the clause "if ZFC is consistent"; it's obviously consistent because it's true.  (So the consistency of ZFC is one example of truth beyond what is provable in ZFC). He knows that CH either holds or doesn't hold in the world of sets, but he probably doesn't know which way it goes. So he would be happy to find new axioms that are clearly true (as the ZFC axioms are) and decide CH.  

I think I've stated both the formalist and the Platonist positions in fairly extreme forms, so I wouldn't be surprised to get comments from actual formalists and actual Platonists objecting that I've unfairly distorted their positions.  I hope such comments would include corrections of the distortions, so as to build a more accurate picture of the positions that people actually take.
The first type of assertion is well-defined and the second is undefinable.