Spectral invariants defined via Embedded Contact Homology (ECH) or the closely related Periodic Floer Homology (PFH) satisfy a Weyl law: Asymptotically, they recover symplectic volume. This Weyl law has led to striking applications in dynamics (smooth closing lemma) and symplectic geometry (simplicity conjecture). In this talk, I will report on work in progress concerning the subleading asymptotics of symplectic Weyl laws. I will explain the connection to symplectic packing problems and the algebraic structure of groups of Hamiltonian diffeomorphisms and homeomorphisms.