Many people use the terms more or less interchangeably (often using "velocity" when they mean "speed" but want to sound more impressive), but there is a very simple but vital difference between the two.
The easiest way to explain the difference between speed and velocity is that speed is a number, while velocity is a vector. Speed is the magnitude of velocity.
Speed has no direction associated with it; it's just a value, like 50 m/s or 60 mph.
But velocity has a direction, and is generally expressed in one of two forms: Components in x, y, and z, like this:
<30 m/s, 40 m/s, 0 m/s>
or magnitude and direction, like this:
50 m/s at 53 degrees north of east.
I've chosen these so they are in fact the same vector. This is analogous to the difference between rectangular (Cartesian) coordinates and polar coordinates.
Speed is the magnitude of direction; the above velocity vector has a magnitude of 50 m/s, so the speed is 50 m/s.