I teach 7th grade students about the difference between speed and velocity. One of them ask me why do physicists create the concept of velocity. I cannot answer. I don't know precisely why do we care about the difference between speed and velocity.
Answer
We care so that we can conveniently calculate motion in 2- or 3-D.
Most motion important to us happens in 2- or 3-D (cars moving over land, airplanes flying through the sky, etc). Vectors make it convenient to handle quantities in more than 1 dimension, so we use vector quantities for position, velocity, and acceleration when describing 2- or 3-D motion.
We care so that we can find out the motion of things being pushed/pulled in different directions.
Not all motion happens along a straight line. Forces do not always push along a single direction. For example, shooting a basketball entails that you push the basketball upward and forward, while gravity pulls it downward. The basketball quite clearly does not move in a straight line; it moves in a curve, constantly changing direction.
In cases like these, we use vectors to describe motion. Vectors make it convenient to handle quantities going in different directions, because they were designed precisely to handle directions!
This is why we have the concept of a vector velocity (as well as position and acceleration): to handle motion where different directions are involved.
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