Abstract tangent functors

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As physicists explore aspects of space and time, it is important for them to have mathematical structures to provide a framework for their calculations. Until recently, the standard mathematical structure for many physical objects was the structure known as a smooth manifold, and the abstract study of smooth manifolds became known as differential geometry. But new objects in physics have forced mathematicians and physicists to realize that smooth manifolds were not enough. Recent researchers have moved beyond smooth manifolds, developing a wide array of more complicated mathematical structures such as diffeological spaces, Fermat spaces, convenient vector spaces, and models of synthetic differential geometry. One pressing question, then, is to understand how to evaluate and compare these different extensions of differential geometry. When should researchers in mathematics or physics use diffeological spaces, say, as opposed to convenient vector spaces?