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The relation of the time evolution operator $\mathrm{e}^{-\mathrm{i}Ht}$ is simply that given any state $\lvert\psi\rangle$ the time-dependent state $$ \lvert \psi(t)\rangle = \mathrm{e}^{-\mathrm{i}Ht}\lvert \psi\rangle$$ solves the time-dependent Schrödinger equation $$ \mathrm{i}\partial_t\lvert\psi(t)\rangle = H\lvert\psi\rangle.$$

Your solution by separation of variables implicitly assumes that your state is an eigenstate of the Hamiltonian with eigenvalue $E$, or equivalently an eigenstate of the time evolution operator with eigenvalue $\mathrm{e}^{-\mathrm{i}Et}$. These eigenstates, under suitable assumptions, form a basis of the state of spaces but not every solution to the Schrödinger equation is an eigenstate: It's a linear equation, so the sum of solutions is still a solution, but the sum of two solutions with different constants $E_1,E_2$ has no constant associated to it anymore.