The tensor product of matrices combines two matrices into a larger matrix, resulting in a multidimensional array that captures interactions between the two. In contrast, the outer product produces a matrix from two vectors, yielding a rank-one matrix that represents their pairwise products. While both operations involve combining matrices or vectors, the tensor product is more general and can handle higher dimensions, whereas the outer product is specifically for vectors. The discussions on Mathematics Stack Exchange provide further insights and examples to clarify these concepts. Understanding these differences is crucial for applications in linear algebra and quantum mechanics.