Take, for example, NDSolve. The Documentation Center page for that asserts: "With the default setting Method->Automatic, the Wolfram Language will automatically try to pick the best method for a particular computation."
This strongly suggests that Method -> Automatic does pick one and the same method for all differential equations, but instead that the method may depend on the form of the equation(s), tests to see whether a stiff system is involved, etc., and also that the method is changed dynamically as the integration proceeds.
In fact, if you look at the SomeNotesOnInternalImplementation page cited in a previous answer, you'll read that, for ODEs, NDSolve "by default uses an LSODA approach, switching between a non-stiff Adams method and a stiff Gear backward differentiation formula method."
With Mathematica being a proprietary product, I hardly expect Wolfram Research to reveal much more than that!