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There are a number of advantages to working with tetrahedral meshes. They are a simple 3D extension of triangle meshes, the dominant representation in computer graphics.

Tetrahedral meshes are naturally adaptive, concentrating most of the tetrahedra in the interesting parts of the model. Also re-meshing operations such as simplification and subdivision have been well-researched.

With a tetrahedral mesh there is a simple correlation between surface and volume. The boundary faces of a tetrahedral model form a watertight triangle mesh that is perfect for both interactive and offline rendering.

Tetrahedral elements conform to local geometry to represent the shape with arbitrary precision. In the top image, I have drawn the edges separating different material types in green, and the remaining edges in purple.

Finally, many popular simulation techniques, such as FEM, have been designed to work with tetrahedral models.