Technically birational geometry is local geometry of algebraic varieties, yet it feels completely different from local differential geometry, which is more or less trivial. Is there some subtle similarity between them?
To think of birational geometry as "local geometry" of varieties is pretty misleading. As Hurkyl says in a comment, it is about the generic behaviour of varieties.
But actually, it is about something a bit more precise than that. It includes the theory of resolution of singularities (any variety is birational to a smooth one), and the theory of minimal models (whose goal is to find canonical members of each birational equivalence class), and these two theories are to a large extent what birational geometry is about. (Another important topic is to find normal forms for birational maps.)
A good example to focus on is that of surfaces (the case of curves is a bit misleading, since for smooth projective curves, being birational is equivalent to being isomorphic):
With a couple of exceptions, each birational equivalence class contains a minimal model, unique up to isomorphism, such that any member of the class admits a birational morphism to the minimal model (so a birational map that is actually defined everywhere). Furthermore, the minimal model has a concrete characterization (it contains no $-1$ curves). Finally, any birational map between two members of the birational equivalence class can be factored as the composition of blowings up and blowings down.
The closest relationship to differential geometry I know is a program (due to Song and Tian, and probably others too) to recover the minimal model classification in terms of the existence of various kinds of metrics on varieties. (One applies various flows such as Khaler--Ricci flow; does appropriate surgeries to pass through singularities of the flow; and eventually ends up with the minimal model. So the blowings down that are required to get to the minimal model are achieved by the surgeries.) I think this program is more global than local in nature (e.g. the obstructions to the existence of certain kinds of metrics on varieties are global), although this is far from my area of expertise, so I can't say anything very definitive. Here is one relevant paper; backtracking through its list of references will lead to others.