Measured voltage and current different from the calculated value
The explanation is pretty simple. The forward voltage estimate you used for the LED is wrong. Instead of being 3V at 20 mA, it is 3.57V at 14mA. You will get pretty close to what you want if you just re-calculate using 3.6V instead of 3V.
So, using your equation but with the modified forward voltage we have:
R = (5-3.6)/0.02 = 70 Ohms.
Standard values that might work are 75 Ohms, or 71.5 Ohms or 69.8 Ohms. If you run it at 20mA, make sure it will not burn out. Often 20mA is the max current for an LED, and is much brighter than necessary. Using 150 Ohms at 14mA might be better in practice.