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: 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.
A: A mathematical description of the differences between speed and velocity is simple enough, and definitely essential; this has been elaborated upon by some of the answers here already.   As far as the question "why should we care" goes, or how to get 7th graders interested well...more reasons than I can probably think of.  Try something like sports.  A quarterback can throw a football fast (speed), but its not going to do any good unless he throws it to the right place (trajectory and or velocity)...  Or driving.  You can drive the speed limit all you desire, but you will still get a ticket (or worse) if you are driving down the wrong side of the road (trajectory, velocity).  Physicist care about these and their differences for similar reasons, but perhaps applied to a wide variety of different scenarios.  Try getting them to think of some scenarios of their own by their own imaginations that they can apply them to, and explain the differences perhaps? 
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: Speed is a scalar, velocity is a vector.  We care because it gives us more information.
In three dimensional space velocity is expressed with three numbers, the speed in each of the three dimensions.
A: Velocity includes information about direction as well as speed (magnitude of velocity). Suppose you are driving at a speed of 50 mph on a road with a washed out bridge, would you not also appreciate knowing your velocity? If you only know your speed you would not know whether you were driving toward or away from the washed out bridge.  To make the distinction clear to your students, you might point out that a steering wheel is a device for altering velocity while maintaining constant speed.
A: If two trains approach each other from opposite directions, on the same train track, they need to calculate their "relative velocities" in order to determine when to stop before they collide. Of course, they would need to know their accelerations too, but if you're explaining to children, this might help. In this case, you can't only consider the "speed" of one train, you need to take into account their directions and each others' velocities (or magnitude of speed, and collude that with direction) to know when they would cross or meet at a certain point. This specific scenario, or others similar in nature, cannot be handled with only speed but require velocities, or rather, relative velocities.
