They're two attempts to measure the same thing, but with different results. Heat index is measured assuming a human walking at 3mph in the shade and attempts to show how hot the air feels when you take into account the humidity (and associated lack of cooling from perspiration).

Wet-bulb is the measurement of the lowest possible temperature a human body can achieve under current ambient temperatures and humidity (taking into account current wind speed). Basically the air temperature minus the ability to cool yourself by sweating given the current wind speed.

They're both designed to show how hot the air feels to your body rather than just air temperature, just with different calculations and assumptions.