Is SR and GR Time Dilation the same thing?

[tom.stoer]

The purely mathematical parts of both SR and GR define a function \tau that takes piecewise smooth timelike curves to positive real numbers. The number \tau(C) is called the "proper time" of the curve C.

A real-world physical clock that moves in a way that's represented by a piecewise smooth timelike curve C in the purely mathematical part of the theory, will display a number at the end of its real-world physical journey and another at the start of it. The difference between those numbers is \tau(C)[/color].

Now, the purely mathematical parts of SR and GR don't say that. They just associate the term "proper time" with the function \tau. So we need to consider the preceding paragraph a part of the definition of each of these two theories.

Let me know if this is still unclear.

That's exactly what I was saying in my first post

In a sense they are the same thing. Proper time along a curve C in spacetime is calculated according to

\tau = \int_C d\tau

Now compare two curves C and C' both connecting two points A and B in spacetime, and calculate the difference for proper times tau and tau' measured along C and C', respectively

\Delta\tau_{C,C^\prime} = \Delta\tau_{C_{A\to B}, C^\prime_{A\to B}} = \int_{C_{A\to B}} d\tau - \int_{C^\prime_{A\to B}} d\tau

All these formulas are valid for both SR and GR and for arbitrary timelike curves. The difference arises only when looking at specific curves i.e. specific experiments
case 1) a geodesic C ('twin on earth') and a curve C' deformed by acceleration ('the twin in the spaceship')
case 2) a geodesic C ('a satellite orbiting the earth') and a curve C' with non-constant radius measuring the difference in gravitational potential

The difference is that in SR we are asking special questions, whereas in GR we are allowed to ask more general questions. The mathemical difference is that in SR the underlying manifold on which the curve is defined is restricted to a flat manifold [which allows for a metric which is globally diag(+1, -1, -1, -1)] whereas in GR the manifold can be any Riemannian manifold.

In that sense the time dilation in SR is nothing else but the effect of an arbitrary curve on a fixed, flat manifold, whereas in GR time dilation is due to arbitrary curves on arbitrary manifolds - for which disentangling effects due to the curve itself and due to the manifold is no longer possible.

Remark: I guess one source of confusion is that quite often time dilation in SR is explained w/o restricting the two curves to intersect at a common end point. I think that in general cases in GR this is no longer allowed, the two curves C and C' must connect two points A and B in spacetime.