Is the reactivity of Benzene with an -N(R)₃⁺ group greater than its reactivity with an -NH₃⁺ group? Both are m-directors, and Electron Withdrawing groups, but which one is more reactive? At first sight, one might think clearly $\ce{-N(R)_3^+}$ is more reactive, but when you factor in that $\ce R$ is actually an o,p-director, hence activator, then I think $\ce{-N(R)_3^+}$ is indeed stronger than $\ce{ NH_3^+}$. I can't seem to find any verification on the internet for this.
$\ce{Ph-NR_3^+}$  is likely weaker than $\ce{Ph-NH_3^+}$ because the mechanism of deactivation is based on the amount of positive charge localized on the Nitrogen:  Alkylgroups will stabilized the positive charge through sigma donation.

That being said, the difference will be small.  Additionally, there are deprotonation side-reactions to consider for anilium ($\ce{Ph-NH_3^+}$) ions, which may compromise any discussion of deactivation.