One often hears that Special Relativity is a "Spacetime Theory". Indeed, this is the predominant way to view the theory, and has been since Minkowski's famous pronouncements on the subject.
Certainly the spacetime framework is a very elegant one, and summarizes very concisely and graphically the results of the theory. But I want to emphasize that one word very clearly: results. The spacetime framework gives us a good way to visualize what the theory predicts, but it gives us little or no understanding of why the theory predicts such things.
For example, a moving object contracts. Why? In the spacetime paradigm this is "explained" by the differences in coordinate systems used by the two observers, and particularly by their different definitions of simultaneity.
But this is rather circular. Coordinate systems create the appearance of contraction, but what creates the coordinate systems? Well...obviously the observers create them themselves, my measuring things out with their own rulers (and clocks). So actually we need to understand the rulers first, before we can understand the coordinate systems, and not vice versa.
Let me give a specific problem that is hard from the spacetime viewpoint. Consider a spaceship which is accelerating constantly, moving faster and faster. We know that it will be contracting; but exactly how does this happen? Does the nose contract towards the tail, or vice versa, or do both contract towards a point in the center? The question does have a definite answer, because both the nose and tail of the ship have a definite trajectory, fully predictable by physics. But I challenge anyone to produce this answer by drawing spacetime diagrams, or computing Lorentz transformations.
I will give my own answer in a future post. For now I will only point out that, in reality, the contraction of a moving object is caused by changes to its internal forces and fields, most notably the electromagnetic field. Understanding this, one can tackle the problem and it is not particularly hard. One also gets past the circularity described above, because one sees that moving rulers (and clocks) are altered by concrete physical mechanisms, so that observers measuring things with them will naturally build different coordinate systems using them.
The energy/mass relation is also quite mysterious from the spacetime viewpoint. Consider this simple scenario: an electron and proton come together to form a hydrogen atom. This process gives off light, hence the atom has less energy than the electron and proton did separately, hence the atom has less mass than the separate electron plus proton. But why? Why is it harder to accelerate an electron an proton bound into an atom, than to accelerate them when separated? I have no idea how to address this question within the spacetime viewpoint, but it is quite simple if one thinks in terms of the physical mechanisms which give rise to the mass/energy formula.
I discuss these sort of things in more detail in my new book, Relativity Made Real (www.relativitymadereal.com). Indeed, these sorts of questions are the reason I wrote the book (although I don't explicitly answer the first one, because it is a bit too in-depth for a popular book).