From my naive understanding, an outer product appears to be a particular case of a tensor product, but applied to the "simple" case of tensor products of vectors. However, in quantum mechanics there appears to be a distinction between the two. For example, given two state vectors ("kets"), $\lvert\psi\rangle$ and $\lvert\phi\rangle$, an outer product of the two is given by $$\lvert\psi\rangle\langle\phi\rvert$$ and this corresponds to a representation of some operator acting on vectors in this Hilbert space.
However, if we consider the two kets, $\lvert\psi\rangle$ and $\lvert\phi\rangle$ as representing two single-particle states, then the tensor product of their two single-particle Hilbert spaces gives a two-particle state of the form $$\lvert\psi\rangle\otimes\lvert\phi\rangle\equiv\lvert\psi\rangle\lvert\phi\rangle$$ which gives a new two-particle state in the tensor-product Hilbert space (of the two single-particle Hilbert spaces).
Is the distinction that the outer product maps two vectors in a given Hilbert space to an operator acting on that same Hilbert space, whereas, the tensor product maps two vectors from two different Hilbert spaces to a vector in the (tensor) product Hilbert space?