The definition of the ‘Q’ source is the source of the material found in both Matthew and Luke, but not in Mark. But while the similarities constitute the substance of Q, the differences between Matthew and Luke provide the strongest argument for the existence of Q.
If it were simply a question of similarities, one could account for them reasonably in any number of ways – as evidence that Matthew used Luke (or vice versa), that both knew a common written source, that both knew the same oral traditions, that both heard Jesus say the same things, and so on.
It is the differences that make it unlikely that Matthew used Luke or vice versa. It is hard, on the supposition of such a literary link, to understand how they could end up with incompatible genealogies, incompatible infancy narratives, and incompatible accounts of the death of Judas. Differences and alterations could certainly be explained in terms of the hypothesis of direct literary dependence. But agreement on large segments, with variations that make sense in terms of the alteration of a saying for particular reasons, and yet disagreement on narrative and geneological details without any obvious reason for those differences, suggests that we are not dealing with direct literary dependence and redactional alterations.Of course, indirect knowledge of one Gospel by the author of the other, mediated either by memory or by oral transmission, is a possibility that deserves consideration. The differences seem to definitively exclude only one possibility: that the author of either Matthew or Luke had the other’s Gospel open in front of him and used it as a source.
Perhaps on another occasion I’ll discuss the difference Q makes. For now, I simply wished to point out that the difference makes Q.