  This paper consists of two parts. In the first part, we use symplectic
homology to distinguish the contact structures on the Brieskorn manifolds
$\Sigma(2l,2,2,2)$, which contact homology cannot distinguish. This answers a
question from [22].
  In the second part, we prove the existence of infinitely many exotic but
homotopically trivial exotic contact structures on $S^7$, distinguished by the
mean Euler characteristic of $S^1$-equivariant symplectic homology. Apart from
various connected sum constructions, these contact structures can be taken from
the Brieskorn manifolds $\Sigma(78k+1,13,6,3,3)$. We end with some
considerations about extending this result to higher dimensions.

Symplectic Geometry