Photoluminescence has been used to find out the band gap of the material. However, the value of band gap calculated from photoluminescence comes out to be slightly higher than calculated from optical absorption data. No more optical property can be calculated from photoluminescence. PL intensity is affected if the recombination centers are large. Hence, one can use this experiment to know the defects in the material.
Actually I would say that ther's a lot of information that can be obtained from photoluminescence spectroscopy in addition to the band gap. Depending on the system you are considering, for example you can get some insight into excitonic energies and different recombination mechanisms. You can study the PL emission in function of the polarization of the incident laser or you can see how the recombination energies depend on the temperature. In QDs you can study phenomena such as photodarkening etc.
For what concern the optical sphere I would say that it's not necessarily the best setup one can use. Depending on the type of study one wants to carry on, a µPL setup with the possibility of performing PL measurements at cryogenic temperature could be better for example...
PL can give lot of useful information about the material as indicated above by Lorenzo. However, it is not a reliable technique for getting the bandgap, although often used. Taking GaAs as an example, there are many impurity related emission lines within a few tens of meV from the bandgap, and your PL is usually coming from these impurities. However, if you do not require a high accuracy, you can still get an estimate for the bandgap from the PL. Furthermore, strong emission can even come from much deep impurities or defects (e.g., hundreds of meV below the bandgap), then bandgap from the PL would be very wrong. Therefore, for a new material, it is very unreliable to use PL to get the bandgap. I would say transmission or absorption is probably the most reliable technique. Otherwise, reflectance is better than PL.
Obviously, the first thing you can learn from PL will be if the material is an efficient emitter under optical excitation. Of course you would need some reference or experience with other materials. Plotting the intensity versus excitation power, you will be able to know how the efficiency changes with excitation level; and further possibly the internal quantum efficiency (see this example: Article Comparative studies of optoelectrical properties of prominen...
). Also, from how the peak position shift with the power, you could also get some idea of the nature of the emission (e.g., band edge vs. defect/impurity).
If some polarizers are used, you can get some idea of the symmetry of the system, the information about the spin of the states, etc.
The list is very long, because any changes in material properties could potentially have some effects in PL, either significantly, subtly or weakly. PL alone usually is not enough to give us a definite answer to any question, in particular those measurements using one power, one temperature.