There is a lot of confusion in this thread, and it seems like I caused some of it, so I'll jump in.
It is important to deconstruct the layers of structure we're using in GR:
1. Topology. Topology cares only about connectedness and continuous maps between spaces. Manifolds are topological objects. For example, flat space has the base manifold R^4, and the Reissner-Nordstrom geometry has the base manifold S^2 x R^2.
A sphere and a cube are the same as topological manifolds. They are both the manifold S^2.
Topologically speaking, manifolds can be given an atlas of charts; that is, a set of continuous maps of open regions of the manifold to R^n, with continuous transition functions between them, such that the whole manifold is covered by the set of maps.
2. Differential structure. This gives a definition of "smoothness" on manifolds. Now the atlas of charts is required to be smooth, with smooth transition functions.
As
differentiable manifolds, the sphere and the cube are
not equivalent, because the cube has 8 points where it is not smooth.
Also, the (maximally-extended) Schwarzschild and Reissner-Nordstrom geometries are
not equivalent as smooth manifolds (despite the fact that they are both S^2 x R^2 topologically), because Schwarzschild has two singularities and Reissner-Nordstrom has a countably infinite number of them.
3. (Pseudo)-Riemannian structure. This is where we give the manifold a local notion of distance and angle; i.e., a metric tensor.
Now, there are three kinds of maps between manifolds worth talking about, depending on how many layers of this structure are preserved:
Homeomorphisms preserve only layer 1. The sphere and the cube are homeomorphic.
Diffeomorphisms preserve up to layer 2. The sphere and the cube are not diffeomorphic; however, the sphere
is diffeomorphic to any topological sphere that is stretched and distorted in any desired way, so long as it is still everywhere smooth. Diffeomorphisms do not care about size or shape.
Metric-preserving diffeomorphisms, which are maps \varphi : (M, g) \rightarrow (N, h), smooth with smooth inverse, such that g = \varphi^* h (i.e. the metric on M is the pullback along \varphi of the metric on N). Metric-preserving diffeomorphisms
do care about size and shape, and are fully equivalent to coordinate transformations.
There is one claim I made in a previous thread which is wrong: I claimed that a sphere of radius A is not diffeomorphic to a sphere of radius B. Clearly this is wrong, because the smooth structure comes prior to the metric structure, and diffeomorphisms care only about the smooth structure. Any two round spheres, of any radius, are diffeomorphic. A sphere is also diffeomorphic to a sphere with a smooth bump on it, etc.
What
is true is that a sphere of radius A and a sphere of radius B fail to be "metric-preserving diffeomorphic" (which unfortunately does not have a convenient word to describe it). This is because there is no diffeomorphism between them such that the metric on sphere A is the pullback of the one on sphere B. Hence the "metric-preserving" condition makes things quite rigid.
In general, local conformal transformations g \mapsto e^{2 \varphi} g are diffeomorphisms, provided that e^{2 \varphi} is everywhere finite and strictly positive. If these conditions are broken, then a local conformal transformation can change the topology, and thus fails to be even a homeomorphism (for example, all 2-dimensional orientable manifolds are locally conformal to each other).
Finally, there is the question of what kinds of transformations leave Einstein's equations invariant? First, look at vacuum solutions with cosmological constant:
R_{\mu\nu} - \frac12 R g_{\mu\nu} + \Lambda g_{\mu\nu} = 0.
Under conformal rescaling by a
constant, g \mapsto a^2 g, we have R_{\mu\nu} \mapsto R_{\mu\nu} and R \mapsto R/a^2, and hence the equation is invariant if we also assume \Lambda \mapsto \Lambda/a^2. What about more general conformal rescaling? From here:
http://en.wikipedia.org/wiki/Ricci_curvature#Behavior_under_conformal_rescaling
one has for g \mapsto e^{2 \varphi} g
\tilde{\operatorname{Ric}}=\operatorname{Ric}+(2-n)[ \nabla d\varphi-d\varphi\otimes d\varphi]+[\Delta \varphi -(n-2)\|d\varphi\|^2],
which indicates that Einstein's equation cannot be invariant under general such transformations. Therefore it is clear that
general diffeomorphisms are not a symmetry of Einstein's equations!
One could consider moving the extra terms to the right-hand-side and treating them as matter sources, but in a sense, the equation with sources
R_{\mu\nu} - \frac12 R g_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{1}{8 \pi G} T_{\mu\nu}
is
trivial; we are simply taking some combination of curvatures and calling it "T_{\mu\nu}". We are only really interested in the kinds of T_{\mu\nu} that describe physically-reasonable distributions of matter, so to call it an "'invariance" of Einstein's equations where T_{\mu\nu} can be arbitrarily modified is kind of silly.
Therefore I conclude that Einstein's equations are not invariant under
all diffeomorphisms, but only under certain kinds. Certainly the metric-preserving diffeomorphisms are included, since they are equivalent to coordinate transformations. I suspect that global rescaling by a constant is the only other possibility.
In any case, this means the whole argument over "active" vs. "passive" diffeomorphisms is a moot point. The only maps that need considering are the maps that are equivalent to coordinate transformations. And so GR's "diffeomorphism invariance" is really a trivial fact;
any theory can be written in a parametrization-invariant way.
