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$S^2$ is 2-dim sphere , $T^2$ is 2-dim torus. $L^2$ is Lebesgue spaces. What is the difference of $L^2(S^2)$ and $L^2(T^2)$ ?

In fact, I don't know how to define the difference ,because isomorphism of Hilbert space can't distinguish them. But I feel there should be some difference.

This is a loose question, I am sorry for this , but I really don't know how to ask it exactly.

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    Well, one space contains (equivalence classes of) functions on $S^2$ and the other contains functions on $T^2$... I don't really understand what you're asking. It is true that the spaces are isometric which is sometimes a useful thing to know but it doesn't mean that they are the same... you could have asked "What is the difference between $\mathbb{R}^3$ and the vector space of real-valued polynomials of degree $\leq 2$?" and the answer would have been the same: "They are different (the underlying sets are different, the addition is defined differently, etc), but are linearly isomorphic." – 2017-02-23
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    @levap So, what is the difference of functions on $S^2$ and $T^2$ ? how to describe the difference , and how the difference will be drown when put them into $L^2$ ? In fact, I am not clear to my question. Sorry about this vague question. – 2017-02-23
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    Somehow, you should not even compare them! It is like asking what is the difference between $L^2$ of the unit segment vs whole R. They are different function spaces. – 2017-02-23

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This doesn't apply to Hilbert spaces, but you might be interested in the fact that if you shift your attention to Banach algebras you have the Gelfand Representation, which I think of as a way to go from a function/operator space back to the underlying topological space. I think this addresses the underlying spirit of your 'loose' question.

If that scratches your itch, you could look into Noncommutative Geometry, which takes the ideas further.

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Naturally, the first Hilbert space is a representation of $SO(3)$, and splits explicitely as an orthonormal Hilbert sum of finite dimensional irreducible representation of $SO(3)$ (theory of spherical harmonics), whereas the second is a representation of the abelian group $R/Z^2$ and splits as the direct sum of (one dimensional) representation of this group (Fourier series).

But of course as separable Hilbert spaces they are isomorphic.