I have been training with Power since 2002 when I purchased my first SRM. Before power I had been training with heart rate since 1990.

Power offers a new dimension to training as it is a totally objective measure of your effort unlike heart rate which changes each day even though the power output may be identical.  If you performed a 20 minute interval for 5 days in a row at the same power output, you would find that your heart rate will most likely drop a beat or two each day, even though the power remains (effort) identical. This drop in heart rate is due to fatigue.

Heart rate is not as accurate when performing high intensity intervals compared with power. When starting an interval heart rate is slow to respond to an increase in intensity which means it may take you 2-5 minutes to elevate your heart rate up to the required higher intensity zone, even though the power output is in zone from the first second. 

There is a difference to how an interval is performed with heart rate compared to power. A typical interval performed with power will show a steady power graph while heart rate slowly rises. A typical interval performed with heart rate will show a spike in power at the start of the interval as the rider brings his heart rate up to the required zone, followed by a steady, slow drop in power as heart rate stays flat (in zone). 

Heart rate training is still a great way to train and for lower intensity training is as good as training with power.  It is also easier to set a limit on intensity (heart rate ceiling) with heart rate compared to power which spikes up with the slightest acceleration.

A big advantage of power is doing regular testing.  Testing shows you your ability on different durations required for performance. Typically, you can test your sprint power (5-10 second), anaerobic power (1 minute), VO2 power (3-5 minutes) and threshold power (10-20 minutes).  Repeating the testing every 8-12 weeks will show completely objectively whether you are improving.

Power makes performance mathematical. With power one can determine what power output it takes to win a time trial or mountain climb and then design a training plan accordingly. Improving your watts/kg (average watts over 20-60 minutes divided by your body weight in kilograms) is how to improve climbing and overall strength. Improving actual power and aerodynamics is the how to improve speed on the flat.

Power has many advantages over heart rate for the cyclist who is serious about improving and who enjoys training precisely and measuring progress. It is the ultimate training tool for the performance cyclists.