Difference between n-dimensional of length m and m-dimensional of length n? With respect to classifying the MNIST digits I've seen the input representation referred to as both a one-dimensional array of 748 elements and a 784-dimensional vector.

Are these perspectives always interchangeable, or is there a material difference between the two?
They are just transposed version.

Often input shape wont matter for informational (performance purposes yes) what will matter is the information inside these vectors and wether its discriminative enough.