Vectors are typically represented by components (units of magnitude in the x,y,z,w,etc. directions), from 1 to n of them. 3D graphics typically deals with 3-4 components per vector. A vector unit can adress all of these components, simultaneously, with one operation. One instruction could operate on multiple data stream elements. Scalar processors, on the other hand, separate individual components to carry out operations, so a cluster of 4 units is needed to operate on a 4-component vector, however when only a simple multiply between 2 scalars or a scalar and a vector are needed, the individual scalar units display greater flexibility (anybody feel free to correct me if I'm wrong).
To the best of my limited understanding, the former proposes a simplistic way of seeing the difference. Someone else could probably jump in and help you out some more.