The Koh-Tindell graph, drawn below, is a graph on 7 vertices that is vertex-transitive but not arc-transitive. It is vertex-transitive since all rotations are automorphisms, but I don't have a proof for the "obvious" fact that these are the only automorphisms. Bondy-Murty ask to
prove that this graph is not arc-transitive, so such a "simple" (not computer-aided brute-force) should exist, say for the fact that there is no automorphism sending an outer arc (of the polygon) to an inner arc (a "diagonal").
The usual method of looking at the degrees of the arcs in the line digraph fails since the digraph is 2-diregular and has many symmetries.