Given the ILC's projected high cost (~$10B) and effort required, capability to study the Higgs boson is an absolute must if it's ever going to be built. Especially the design hinges on what the Higgs mass is, and therefore how much energy will be required. For a low-mass Higgs (e.g. 125 GeV) the original ILC at 500 GeV would be enough. They've also considered a 1 TeV design, as well as a 3 GeV CLIC. People will probably argue for higher energy anyway, enough that supersymmetry will also be within reach.
All of these machines are electron-positron colliders. Proton colliders like the LHC are really quark colliders, and the individual quarks inside the proton have a considerable spread in energy, which limits the energy resolution of your results. But if you're colliding electrons you know the energy more precisely, and that's their big advantage.
The LEP, which previously occupied the site where the LHC is now, was an electron-positron ring collider which operated at energies up to 200 GeV. Such colliders must cope with a large amount of synchrotron radiation - the main difficulty is radiation damage to the instrumentation.