You can store ALL transformations in one matrix, rather than a matrix and a vector. Also, matrices are associative. This means a translation, then a rotation, is not the same as a rotation and then a translation. The translation could be affected by the previous rotations, or scales. This could be useful.
Also, when multiplying a 4*4 matrix with a 4 component vector, we could make the forth component of the vector 0, meaning it'll ignore the translation. A vector with a forth component of 0 means it's normalised. We don't want a normalised vector to me translated.
Yes, you could do all of this without matrices. It's just simpler and faster this way. You especially wouldn't want to manipulate a game with half matrices and half vectors.