Although it seems very reasonable to expect that a smaller
set must have strictly fewer subsets, which is another way
of stating your property, in fact this property is
independent of ZFC. Many people find this surprising.

(And for this reason I mentioned it in [this
answer](http://mathoverflow.net/questions/1924/what-are-some-reasonable-sounding-statements-that-are-independent-of-zfc/#6594)
to the MO question requesting examples of
reasonable-sounding statements that are independent of
ZFC.)

You are asking whether the continuum function
$\kappa\mapsto 2^\kappa$ is injective, and the fact is that
if ZFC is consistent, then this assertion is neither
provable nor refutable in ZFC.

On the one hand, the property is relatively consistent with
ZFC, as you observe, since it follows easily from the GCH.
On the other hand, in Cohen's original model of
$\text{ZFC}+\neg\text{CH}$, obtained with his method of
[forcing](http://en.wikipedia.org/wiki/Cohen_forcing), he
forced over a model of GCH to add $\omega_2$ many Cohen
reals, and the result was $2^\omega=2^{\omega_1}=\omega_2$,
which violates your property. This is usually one of the
first nontrivial forcing arguments that set-theorists
learn.

It is also a consequence of [Martin's
Axiom](http://en.wikipedia.org/wiki/Martin%27s_axiom) that
$2^\kappa=2^\omega$ for all $\kappa\lt\frak{c}$, and if one
has MA plus $\neg\text{CH}$, which is known to be
relatively consistent by the forcing method, then there are
again counterexamples to the requested property.

Meanwhile, one can show by forcing that the injectivity of
the continuum function is not equivalent to GCH, since by
[Easton's
theorem](http://en.wikipedia.org/wiki/Easton's_theorem),
one can find a forcing extension (of any model of GCH) in
which $2^{\aleph_n}=\aleph_{n+2}$ for every natural number
$n$, and otherwise the GCH holds. Such a model exhibits the
desired injectivity property, but does not satisfy GCH. One
can use Easton's theorem more generally to make even more
extravagant violations of GCH, while still ensuring an
injective continuum function.