Definitions
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 n. A differentiable homeomorphism (with differentiable inverse) between differentiable manifolds.
Etymologies
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Examples

The Hamiltonian constraint NH = 0 defines a diffeomorphism structure on spatial manifolds by the lapse function N, but this is not identical to time.
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The main problem for the latter is the general covariance of the field equations of General Relativity: any spacetime model and its image under a diffeomorphism (a infinitely differentiable, oneone and onto mapping of the model to itself) are in all observable respects equivalent to one another; all physical properties are expressed in terms of generally covariant relationships between geometrical objects.

Other: It's not nonperturbative quantization of diffeomorphisminvariant gauge theory.

For mathematical reasons not relevant here, it turns out to be possible to take a given model spacetime and perform a mathematical operation called a “hole diffeomorphism” h* on it; the diffeomorphism's effect is to shift around the matter content

Figure 5: “Hole” diffeomorphism shifts contents of spacetime

Maybe this is the resolution for the representation facts which you refer to from CFT, insofar as those facts perhaps pertain to continuous (projective) representations of the diffeomorphism group of the circle.

Nontrivial unitary lowestenergy reps of diffeomorphism and gauge algebras are necessarily anomalous, and a quantized observer jumps right out of their representation theory.

As is well known already in 1D, interesting lowestenergy representations of the diffeomorphism algebra are necessarily anomalous (Virasoro algebra), and the higherdimensional case is not different (multidimensional Virasoro algebra).

In AdS/CFT, not only the bulk spacetime but also the bulk diffeomorphism group are emergent: the CFT fields are completely invariant under the bulk diffeomorphisms (this is also what happens in the much more common phenomenon of emergent gauge symmetry).

Manifest diffeomorphism invariance seems too limited if we want to describe string theory.
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