Is Teleparallel Gravity really equivalent to General Relativity?

An axiomatization of the so-called Teleparallel Equivalent to General Relativity is presented. A set of formal and semantic postulates are elaborated from where the physical meaning of various key concepts of the theory are clarified. These concepts include those of inertia, Lorentz and diffeomorphism invariance, and reference frame. It is shown that Teleparallel Gravity admits a wider representation of space-time than General Relativity, allowing to define properties of the gravitational field such as energy and momentum that are usually considered problematic. In this sense, although the dynamical equations of both theories are equivalent, their inequivalence from a physical point of view is demonstrated. Finally, the axiomatic formulation is used to compare Teleparallel Gravity with other theories of gravity based on absolute parallelism such as non-local and f(T) gravity.