Why won't a LED bulb work in a standard light fixture?
I found the answer! @diceless set me on the right track in looking at where the foot of the bulb is connecting to the fixture.

Since it was an older fixture (installed in the 80's, I think), the hot contact (Number 1 in this diagram ) was bent down, and it was connecting with the smaller bulb but not the bigger LED bulb.

With the circuit turned off at the breaker I reached in with a wooden tongue depressor and bent it upwards slightly. Then, I made sure to screw the LED bulb in tightly. Flicking the switch it was bright as day!