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In a tetrahedron all the triangles have to be congruent. In a Triangular Pyramid, the lateral faces are all congruent to each other, but not always with the base triangle.

Not necessarily. The lateral faces need to be congruent if it is a right pyramid, not otherwise. The apex of the pyramid need not be directly above the centre (however defined) of the base. Think of a triangular pyramid doing an impersonation of the Leaning Tower of Pisa!

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