Let $R$ be the complete local ring of a complex plane curve germ and $S$ its normalization. We propose a conjecture relating the virtual weight polynomials of the Hilbert schemes of $R$ to those of the Quot schemes that parametrize $R$-submodules of $S$. We prove an identity relating the Quot side to strata in a lattice quotient of a compactified Picard scheme, showing that our conjecture generalizes a conjecture of Cherednik's beyond the unibranch case, and that it would relate the perverse filtration on the cohomology of the Picard side to the stratification. We also lift our work to a parabolic refinement where we track partial flags. We propose a Quot version of the Oblomkov-Rasmussen-Shende conjecture, relating the parabolic Quot side to Khovanov-Rozansky link homology. It becomes equivalent to the original Hilbert version under our Hilb-vs-Quot conjecture, but is more tractable. For germs of the form $y^n = x^d$, where $n$ is either coprime to or divides $d$, we prove our Quot version in its full form. No similar result keeping all three gradings is known for the Hilbert version. Finally, we enhance the Quot version to incorporate a polynomial action on the link homology, as well as its $y$-ification; neither has a Hilbert analogue.