In the existing literatures, some authors labelling X-axis as Raman shift and others as wavenumber for the Raman plot. Which will be the appropriate to use. Is their any difference between these two terms?
You have either wavenumber for mechanical oscillations in system, or difference between two optical wavenumbers for incident and scattered light, that is equal to mechanical one and defined as Raman shift.
The appropriate (and most common) units for the X-axis of a Raman spectrum are the wavenumbers, indeed. The axis TITLE is usually 'Raman Shift (cm-1)', i.e., Raman shift in wavenumbers.
In some exotic case you may see the Raman shift measured in meV (milli electron volts), e.g., in electronic Raman scattering for shallow-donor electrons in semiconductors. This is stipulated by the fact that the donor binding energies are measured in meV (or eV).
Labeling the X-axis with 'Wavenumbers' only would be wrong, since this assumes ABSOLUTE wavenumbers. These are commonly used in FTIR, for example.
It is absolute wavenumbers for mechanical oscillations (like in FTIR spectra), so the Wavenumber is correct description, that is recommended, for example, in Journal of Raman Spectroscopy as preferred units.