What is the difference between Raman scattering and fluorescence?

I think that it is important to recognize the practical difference between Raman scattering and fluorescence.

If the energy of the photon is resonant with some molecular transition (meaning that it is equal to the energy difference between ground energy state and one of the excited states of the molecule), that the molecule can absorb this photon undergoing energy transition. Now, there are many things that can happen with the molecule in this excited state: it can lose some or all of its extra energy in collision with another molecule, it can fall apart if this extra energy is large enough or - if it avoids these things - it can emit extra energy as a photon which is called fluorescence. The energy of the emitted photon is not necessarily equal to the energy of the absorbed photon as the molecule can end up in a different energy state (for example, having more vibrational energy).

There is a reason that Raman process is called scattering. It is a non-resonant process so the energy of the photon is not important (although higher energies are more efficient in inducing Raman scattering). You can imagine molecule as a small antenna which receives electromagnetic radiation and can re-radiate this. In most cases, the molecule will scatter exactly the same energy - this is called Rayleigh scattering. In few cases, small part of this energy is stored in molecular vibration, or, if molecule is vibrationally excited, it can give this extra energy to the photon. When this happens, scattered light has a shift in wavelength and the process is called Raman scattering.

In contrast to fluorescence, there is no excited state in Raman scattering, therefore the process is almost instantaneous, whereas fluorescence has characteristic lifetime of nanoseconds. So, no matter how one can play with words, these two are very different processes and they totally deserve different names.