One often says a hadron collider like the LHC is used for discovering, while an electron collider is rather used for precision measurements. There are a couple of benefits of a high-energy electron collider:
All electrons have roughly the same energy. One can vary the center of mass energy $\sqrt{s}$ and map out resonances (think $ee\rightarrow Z$ or $J/\psi$). Compare this to the LHC, which is actually in a sense a gluon-gluon collider. The gluons in the protons have wildly varying energies (the probability of a certain energy is given by the parton distribution functions), so you never know with which energy your particles are colliding. You are probing all energies up to 7 TeV at the same time, in a way. Even worse, most of the time your initial state particles have different momenta in beam direction, so your whole system is boosed along the beam. These effects smear out and shift your distributions, e.g. if you want to measure the invariant mass of something. This is the reason that we use transverse variables at hadron colliders - there is no initial boost in transverse direction (perpendicular to the beam line).
Related to the above, in some searches for new particles you can see nice mass edges at a lepton collider. They occur when you have a chain decay, say $\tilde\chi^0_2 \rightarrow \tilde\chi^0_1 + Z \rightarrow \tilde\chi^0_1 + \ell^+ \ell^-$. The invariant mass distribution of the lepton pairs rises up to the point $\Delta m = m(\tilde\chi^0_2) - m(\tilde\chi^0_1)$ where all of the energy from the chain decay is carried by the two leptons and none by the neutralino, and then it drops off suddenly. You can look for mass edges at the LHC too, but they will be more smeared out.
As you said, the collisions are 'cleaner'. There are no jets from proton remnants for example. A lot of backgrounds just don't exist there, and if you are running near the energy of an interesting resonance, you will get a lot of signal events.
However, at the energies of the ILC there will still be a lot of jets produced. Also the ILC will have a pretty high luminosity, so they will also have to deal with multiple simultaneous events. And then, there are entirely new problems. After the electron beam is focused in the collision point, it diverges quickly behind it because the electrons repel each other. The part of the detector nearest to the beam line will be 'blinded' from all the electrons ('beamstrahlung'). If you look at data vs. background plots from ILC simulations, you see that the beam-induced background spikes at high $\eta$, at small angles to the beam. It looks like this can be controlled, but it is still challenging for detector designers and analyzers.
It's very probably the LHC experiments will conclusively observe the Higgs Boson soon. But then we still don't know which Higgs Boson it is - the standard model one, or another one with different properties? In the MSSM (a supersymmetric theory), there are five Higgs bosons, two of which are charged. And what's the exact mass (masses)? We will eventually be able to answer these questions at the LHC with a lot of data, but we can do it more precisely with a special tool like the ILC. The same not only applies for the Higgs, but also for other new physics, like SUSY, extra dimensions, or whatever other thing there is to be discovered.
