Fire place is the opening where the fire is, the hearth. The chimney is the structure that carries the smoke out through the roof, the stack, usually brick / stone built.
A fireplace is exactly what it says, the place where the fire is. The chimney is the pipe leading from the fireplace to the roof to take away the smoke.
The chimney is above the fireplace.
The chimney may be hidden in the wall, or visible, but it carries the smoke up from the fireplace and out of the house.
You can see the chimney on the outside of the house.
A fireplace is the protected area INSIDE a dwelling in which a fire is laid and burns. a chimney is the vent for the fireplace, the vertical, mostly-exterior structure that serves as a channel for the smoke created in the fireplace.
A fireplace : an open recess in a wall of a room, at the base of a chimney
A Chimney: A passage through which smoke and gases escape from a fire or furnace; a flue.