Q:

Why do we care about the difference between speed and velocity?

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.

A:

I give you a different type of answer. In Italian, and in many other languages I suppose, there are not different words for speed and velocity and so there is not any ambiguity among these concepts.
Velocity it's a vector, but you can refer to it's module because there is not any ambiguity, as you do for any vector quantity. (For istance, I'm sure you would not have made the same question for the momentum)
I think in English you stress the difference between the two only because in your language there were two different words (one from Latin and one from Germanic origins) already before the birth of physics.
In my opinion is only a cultural difference.

A:

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.

