The obvious answer is more energy per collision. This is discussed on p21 in this brochure of FAQs: CERN-Brochure-2017-002-Eng, which I found linked in this old question: Why not build a particle accelerator on ground level? What is the shallowest feasible depth to build one?.
Each proton beam flying around the LHC has a maximum design energy of 7 TeV, so when two protons collide the collision energy is 14 TeV. Lead ions have many protons, and together they give an even greater energy: the lead-ion beams have a maximum collision energy of 1150 TeV. Both collision energies have never been reached before in a lab.
But from Accelerator Science: Proton vs. Electron, collisions are not between lead ions so much as two quarks/antiquarks/gluons within two protons or neutrons. Mostly the ions pass through each other.
Is it simply because more massive ions can be accelerated to higher energies per nucleon, as implied on p19?
In a circular accelerator, such as the LHC, heavy particles such as protons (protons are about 2000 times more massive than electrons) have a much lower energy loss per turn through synchrotron radiation than light particles such as electrons. Therefore, in circular accelerators, to obtain the highest-energy collisions it is more effective to accelerate massive particles.
Or is it that you get more nucleons in the beam with ions?
Either way, why not use something heavier like uranium? And why are Future Circular Colliders at higher energies than the LHC planning to use protons?