We provide more characterizations of varieties with a weak difference term and of neutral varieties. We prove that a variety has a (weak) difference term (is neutral) with respect to the TC-commutator iff it has a (weak) difference term (is neutral) with respect to the linear commutator. We show that a variety \v\ is congruence meet semi-distributive i{f}f \v\ is neutral, i{f}f $M_3$ is not a sublattice of \con a, for ${\bf A} \in \v$, i{f}f there is a positive integer $n$ such that $\v \smc \a(\b\o\g)\leq\alpha \beta_n$.