"Brieskorn manifolds became prominent in contact topology after I.\ Ustilovsky used them to prove the existence of infinitely many contact structures on $S^{4m+1}$. After briefly reviewing his result, we will focus on another class of Brieskorn manifolds for which contact homology cannot distinguish the contact structures, whereas (positive) symplectic homology can. The main difficulty of the proof lies in the computation of the differential of symplectic homology. For this, we will use Morse--Bott methods as well as an explicit perturbation, combined with a symmetry of the underlying manifold. If time permits, I will also explain the resulting transversality problem and how to solve it."