It is my (probably incorrect) understanding that tensors are the generalized form of scalars, vectors, matrices, etc. A scalar being a 0 dimensional tensor, a vector being a 1 dimensional tensor, a matrix being a 2 dimensional tensor, and so on.
Tensors have meaningful operations already rigorously defined in mathematical texts; consider multiplication between a vector and a scalar, dot product of two vectors, matrix multiplication, etc.
A multidimensional array is just data ordered over several dimensions, there are no intrinsic operations. So if you're talking about multidimensional arrays that have such operations defined, it's useful to communicate that distinction by using the name "tensor".
In the same way that it's useful to talk about coordinates rather than "1-dimensional array of length <base size> that respects certain invariants".
