Families often assume they must choose between the two. In practice you usually have both, and they handle quite different parts of the week.
The funeral director
Cares for the person who has died, handles the paperwork and registration of the death, arranges the casket, transport, venue, crematorium or cemetery bookings, and the order of service printing.
The celebrant
Creates and leads the ceremony itself: gathering the story, writing the tribute, shaping the order of the service and holding the room on the day.
You can engage a celebrant directly, and many funeral homes are happy to work with the celebrant of your choosing rather than the one on their roster. If a funeral home offers an in-house celebrant, that is a convenience — but the choice of who tells the story remains yours.