What's the advantage of gamma-gamma colliders? What new physics can be done with it?
The strongest physics quest for a $\gamma\gamma$ collider is the production of Higgs bosons as the only particle in the final state, from a very well known initial state.
This is not achieved neither at future lepton colliders (ILC, CLIC, FCC-ee, CEPC) where the typical production by higgstrahlung comes with additional particles in the final state (carrying some uncertainty), and even less at proton colliders (LHC) where the initial state is unknown and there is additional noise from the proton remnants. Off course this didn't prevent its discovery, but makes precision studies much harder.
If the diphoton signal at $750~$GeV observed in the 2015 run of the LHC is confirmed, it would be a very strong push, although reaching that energy with a $\gamma\gamma$ collider is... let's say expensive!