The crystallographic site symmetry identifies the point symmetry of a specific site in three dimensions (x,y,z) within the unit cell. . It determines also the selection rules that allow specific spectroscopic transitions in the solid. Crystallographic and 'spectroscopic' site symmetry: on the contrary, spectroscopy can be useful to tell you what is the correct site symmetry , even when diffraction data are by themselvs not conclusive. A typical case, for example, occurs when you try to differentiaite between the presence or the absence of an inversion centre in the solid.

I am not sure if I understand correctly, but the symmetry in a crystal is given by the 3D arrangement of its components. I suspect the spectroscopic site symmetry is inherent to the sample itself and more similar to non-crystallographic symmetry in a crystal. A crystalline symmetry is "mathematically correct", in the sense that it follows complete integer operations. A non-crystallographic symmetry can be a point group or anything other than that.

What kind of spectroscopy are you referring to?

The crystallographic site symmetry identifies the point symmetry of a specific site (x,y,z) within the unit cell. See also http://reference.iucr.org/dictionary/Site_symmetry. In this respect, it determines also the selection rules that allow specific spectroscopic transitions in the solid - or, equivalently, by probing a specific ion in a crystal lattice you can infer its (average) local site symmetry from the corresponding spectral features, as done for example by Jia et al. J. Phys. Chem. C, 2010, 114, 17905. In this respect, I do not see in principle any difference between crystallographic and 'spectroscopic' site symmetry: on the contrary, spectroscopy can be useful to tell you what is the correct site symmetry (by the occurrence of symmetry-forbidden bands, for example), even when diffraction data are by themselvs not conclusive. A typical case, for example, occurs when you are trying to discriminate between the presence or the absence of an inversion centre in the solid.

The point symmetry of the site of a crystal is the same for both crystallography and spectroscopy. These are shown in International tables of crystallography for each Wykoff positions of the space group. The point symmetry is vital both in crystallography and also in spectroscopy because It is the symmetery that is seen or felt by the the atom at the site.

Consider cubic pyrochlore structure.For cubic systems the point groups allowed are Oh, O, Td, Th and T. For Eu3+ doped phosphors, then according to the point group symmetry no 5D0-7F0 transitions are allowed, for 7F1&7F2 transitions no splitting is allowed, But when we are monitoring the emission spectra normally 7F1 splits up to two, 7F2 splits up to 3. How is it possible/ then what is the site symmetry.

This may simply mean the ideal pyrochlore structure is actually distorted by small amount changing the local site symmetry and thereby causing the forbidden transitions to be observed. In that case it is a very important result and should be published. But before you do that check carefully the group theoretical arguments. It is easy to make mistakes unless you are very careful.

You can consult the useful book "Site symmetry in crystals", Robert A. Evarestov and Vyacheslav P. Smirnov, Springer (1997).

Theoretically, there is no difference. Take into account that Crystallography is a science which makes use of different techniques, some of them could be spectroscopic techniques, other may be scattering techniques. Thus, mathematically there is only one definition of symmetry. On the other hand, a difference may occur between a spectroscopic and a diffraction observation in a crystal (i.e. two different techniques applied in crystallography). Very likely your question refers to this. In this case, one should consider the different response of the material to the different radiation (in particular the time and space domains of the event that will produce an observed signal, which could be very different depending on the type of radiation). Thus, something has apparently a symmetry if sampled with X-rays and a different symmetry if sampled with visible light. Or even there could be a different symmetry between X-ray and neutron diffraction, because X-rays are sensible to the one electron density, whereas neutron to the spin density, that could have different symmetries.

A phenomenon like diffraction is inherently "centrosymmetric" (due to Friedel law), whereas some spectroscopic techniques (for example SHG) are very useful to ascertain non-centrosymmetry. Again, however, it would be incorrect to say that a crystal is centrosymmetric under the X-ray diffraction and non-centrosymmetric under a laser spectroscopy. The crystal is just the same, X-rays may not be able to fully reveal the non centrosymmetry (although in most cases there is no problem also with X-rays, especially if using the correct energy).

Concerning the pyrochlore-structure crystal, I agree with Tapan Chatterji and I'd like to add the ideal pyrochlore structure could be perturbed by low-symmetry random crystal fields caused by point defects (I think there are no charge defects if Eu substitutes for a host ion without charge compensation). We have studied such a phenomenon in our recent work on centrosymmetric crystals: http://link.aps.org/doi/10.1103/PhysRevB.86.134110 (available in my profile)

and of course you should be sure that Eu ions occupy a sigle possition in the crystal)))

What kind of pyrochlore-structure crystals did you study, by the way?