So as to not over-complicate things, here's the jist of it:
In a nutshell; Heart rate is a measurement of any beating/contraction of the heart, no matter how forceful the contraction or how much/little volume of fluid (blood) is shifted.
Pulse rate is a measurement of beating/contraction of the heart which produces enough pressure (ie, is forceful enough/moves enough fluid to the peripheries) for the pulse to be palpable.
Heart rate = ANY heart beat
Pulse rate = ONLY heart beats which are strong enough to be felt way out at your wrist/knee/ankle, etc.
A good example to illustrate this point is a group of conditions called Supra-Ventricular Tachycardias (SVTs). In SVTs (due to lots of varied reasons) your heart beats much faster than it normally would, this is shown as an increased Heart rate. However, due to the pace of the heart's beating, it doesn't have enough time to completely refill with blood like it normally would. So, whenever# the heart beats (in SVTs), only a small amount of blood is pushed out, not nearly enough to create the pressure required for you to feel the pulse. The Pulse rate is therefore going to be much lower than the Heart rate.
#Note: obviously some beats in SVTs generate enough pressure for a palpable pulse, or else there wouldn't be a pulse at all! It's just that there might be two "weak" beats for every "strong" beat, hence the disparity (Pulse Deficit).