Why doesn't running light work when blinker does?

Did you put the right bulb in. Some bulbs have only one filament vs two filaments.

It was supposedly the "exact fit" for my car when I ordered the part

Assuming its the correct bulb, which it probably is or it wouldn't have fit in the socket, you now need to make sure the bulb actually works. Put power to it or use a continuity meter. If its not the bulb, check to see if there's power to the socket. If there's not you need to figure out why. Look for damaged wiring or a coroded connection. If there is power to the socket then figure out why it doesn't light the bulb. Coroded contacts in the socket for example.