What are some examples of compact Kaehler manifolds (in particular, smooth complex projective varieties) that are not isomorphic as complex manifolds, but are isomorphic as symplectic manifolds (with the symplectic structure induced from the Kaehler structure)?

Elliptic curves should be an example, but I can't think of any others. I'm sure there should be lots...

And in the other direction, if I have two compact Kaehler manifolds (or, again, smooth complex projective varieties) that are isomorphic as complex manifolds, then are they necessarily isomorphic as symplectic manifolds?