All the other answers are great, but I feel like this simple thing was left unsaid:
The word identical is unwise to use, because it is unclear whether it means isomorphic or naturally isomorphic.
V and V* and V** are all isomorphic of course (if they are finite dimensional), after all they have the same dimension.
What you want to say is this: V and V** are naturally isomorphic but V and V* are not! The notion of natural isomorphism is defined nicely by Dylan. If you look it up you'll find that what is actually naturally isomorphic is not V with V** but rather the ** itself. That is, the functor of double-dual on the category of vector spaces over the reals is naturally isomorphic to the identity functor.