While speed and velocity are commonly used in everyday terminology interchangeably, they do in fact represent different quantities.  In the study of physics, there is a specific definition for each.  Speed is defined as the total distance traveled with respect to the total time it took to travel that distance.  Velocity is defined as the change in position of an object with respect to the time it took to travel from its starting position to its ending position.  The quantity of speed carries no specified direction or sign.  The quantity of velocity on the other hand does carry a specific direction and algebraic sign, so it is actually possible to have a negative or zero value even if a distance is traveled.

The formula for speed is S = D/t, where S is speed, D is the total distance traveled and t is the time it took to travel that distance.  The formula for velocity is V = (Df – Di)/t, where V is velocity, Df is the final measurement of distance traveled relative to some point, Di is measured relative to the same point and is the initial measurement of distance where travel begins and t is the time about which travel took place.

Let’s consider an example.  You are driving in the mountains, and miss your turn, and so you end up traveling for 12 minutes for a distance of 5 miles around a mountain only to end up at a location you were previously at.  For this 12 minute period, you began and ended at the same geographic location.  What is your average speed and average velocity for this time period?

Your speed, S = D/T = (5miles/12 minutes)(60 minutes/1 hour) = 25 miles/hour

Your velocity, V = (Df – Di)/T, where Df – Di = 0

V = 0 / 12 minutes = 0 miles/hour

From this example it is illustrated that speed and velocity while similar in definition and units are very different quantities.  In this example, if the 5 mile distance had been traveled linearly away from the starting point, then the average speed and average velocity, while still different in term would have the same value of 25 miles/hour.  On the other hand had the driver went around the mountain and ended up at a stopping point somewhere before the starting point (a location geographically negative to the starting point) then the velocity would be negative based on the difference between the starting and stopping points.

Source:

Serway, R. A., Beichner, R. J., & Jewett, J. W. (2000).  Physics For Scientists and

Engineers, Volume 1 (5th Ed.).  Orlando, FL: Saunders College Publishing.