What is the smallest regular map with a Schläfli symbol of the form {p,p} but not self-dual?

It is S4:{6,6}2,3, or its dual S4:{6,6}3,2. The former has a vertex multiplicity of 3 and a face multiplicity of 2; the latter has a vertex multiplicity of 2 and a face multiplicity of 3.

In the same genus there is also S4:{6,6}3,3, which is self-dual, with vertex multiplicity and face multiplicity both 3.

Regular Maps FAQ to which this is one of the answers.