# Difference between the CKM and the PMNS matrix

My question is very simple: is there a fundametal difference between the CKM (responsible for quark mixing) and the PMNS matrix (responsible for lepton mixing)?

The CKM matrix allows (in charged current) mixing between quark families: for example $W^+\rightarrow u\bar{s}$.

While in the leptonic sector, a decay of type $W^+\rightarrow e^+\nu_{\mu}$ is impossible. However, when switching from $SU(2)$ eigenstates to mass eigenstates in the Lagrangian, I do not see why this decay is not allowed.

And the decay $W^+\to e^+\nu_\mu$ is entirely possible if $\nu_\mu$ actually represents a mass eigenstate of the neutrino, much like the convention we use in the case of quarks.
For this reason, because of the very slow neutrino oscillations, we don't actually use the neutrino mass eigenstates in discussions about particle physics processes. By $\nu_\mu$, we really mean the $SU(2)$ partner of the left-handed muon (the middle mass eigenstate in the 3D space of the charged leptons of a given charge), and then the decay to $e^+\nu_\mu$ is strictly prohibited by the $SU(2)$ gauge symmetry.
But the process $W^+\to e^+\nu_2$ where $\nu_2$ is a mass eigenstate of the neutrinos is allowed because $\nu_2$ contains a nonzero addition of $\nu_e$, and it's this process involving $\nu_2$ that is analogous to the flavor-changing processes with quarks (controlled by the Cabibbo angle etc.).