How are the Weyl & Riemann curvature tensors related to the stress energy tensor in GR? Einstein's vacuum equations, that is without matter, allows the possibility of curvature without matter. For instance, we may consider gravitational waves. 
The question is: Is there some link between the Riemann curvature tensor, and/or the Weyl tensor, and some gravitational "physical" quantities (as stress-energy tensor or total energy)? 
Of course, at first glance, there is no covariant gravitational  stress-energy tensor, so it seems there is no relation, but maybe things are more subtle?
 A: 
Is there some link between the Riemann curvature tensor [...] and some gravitational "physical" quantities*

Maybe you could clarify what you want that would qualify as "physical." Curvature is observable, and IMO is physical. Projects like LIGO are designed to detect gravitational waves. Gravity Probe B was a project that accomplished its purpose of essentially verifying GR's predictions of spacetime curvature in the neighborhood of a gravitating, spinning body. In the simplest terms, curvature can be measured by transporting a gyroscope around a closed path. This is essentially what GPB did.

Of course, at first glance, there is no covariant gravitational stress-energy tensor

But that's only a prohibition on defining a local measure of gravitational-wave energy. For example, in an asymptotically flat spacetime, the ADM energy includes energy being radiated away to null infinity by gravitational waves. If LIGO-like projects succeed, they will measure the energy of gravitational waves.
A: The first point to consider is that the Riemann tensor can be expressed in terms of the Weyl tensor and the Ricci Tensor:
$$R_{abcd}=C_{abcd}-g_{a[d}R_{c]b}-g_{b[c}R_{d]a}-\frac{1}{3}Rg_{a[c}g_{d]b}$$
The Ricci tensor is given by Einstein's equation:
$$R_{ab}-\frac{1}{2}g_{ab}R+\Lambda g_{ab}=8\pi T_{ab}$$
Now the Weyl tensor is not specified by the EFE. However, it can not be arbitary as the Riemann tensor must satisfy the Bianchi identities:
$$R_{ab[cd;e]}=0$$
Applying this last condition to the first equation we obtain that the Weyl tensor must  satisfy: $$C^{abcd}_{;d}=J^{abc} $$
where $J^{abc}=R^{c[a;b]}+\frac{1}{6}g^{c[b}R^{;a]}$.
You can find a proof here.
You can now make the interpretation of this field equations as determining that part of the curvature at a point that depends on the matter distribution at other points. 
