I'm going to write this as an "answer", though I think it mostly amounts to a comment on Dave Stewart's answer.
It is not completely clear to me that the statement "the nilpotent variety of $\mathfrak{pgl}_2$ is reduced" is correct when $p=2$. Well, I suppose that more precisely I mean: it isn't clear that the scheme of nilpotent elements should be viewed as reduced.
Taking a basis $x,y,h$ of $\mathfrak{pgl}_2$ (say, in its 3-dimensional representation), one finds that $ah + bx + cy$ is nilpotent just in case $a^2+4bc=0$. Of course, in char. 2 this amounts to $a^2=0$, which suggests that the scheme of nilpotent elements shouldn't be viewed as a reduced subscheme. (if you don't want to write down the matrices, see e.g. Jantzen "Nilpotent Orbits in Representation Theory" $\S$2.7).
I do doubt (?) that this nilpotent scheme is isomorphic to the scheme of unipotent elements of $\operatorname{PGL}_2$, but (assuming that doubt is correct - I didn't think too carefully about it) I think the reason is more complicated than the statement "one is reduced and the other isn't".