The core difference between them lies in one sentence:

Fire-tube boiler: Fire is inside the tubes and water is outside the tubes.

Water-tube boiler: Water is inside the tubes and fire is outside the tubes.

The working principle of the fire-tube boiler:
The flue gas flows inside the pipes, while water surrounds the pipes in the pot shell. The fuel burns in the furnace, generating high-temperature flue gas. Smoke passes through the interiors of one steel pipe after another. There is a large amount of water outside the steel pipe.
Heat is transferred from the flue gas to the pot water through the pipe wall, heating the water, causing it to boil and generating steam.
The features of the fire-tube boiler: The shell is a large container with abundant water and a large steam drum volume. The heating surfaces are just those few smoke tubes.

The working principle of water-tube boilers
Water is inside the pipe, while flue gas scour the pipe outside. Water first enters the heating surface composed of thin tubes (water-cooled wall, convection tube bundle). High-temperature flue gas flows outside the pipes in the furnace and flue, scouring the pipe walls. Heat is transferred from outside the pipe to the water inside. Water is heated and vaporized inside the tubes, turning into a steam-water mixture, which then returns to the drum to separate the steam.
The features of water-tube boilers
All the water is in the thin tubes, and the drum only serves the separation of steam and water. The heating surface can be arranged in many places, featuring high thermal efficiency and strong pressure resistance.