I saw this question on another site. It asked if a pizza cutter could cut fabric. which is not what I ask. A pizza cutter stays stationary wrt the underground while a straight knife moves. In both cases, though you can exert the same pressure. So why not use a straight knife when cutting a pizza?
Well, I can think of the following reasons:
First, it is much easier to exert considerable vertical force on the pizza cutter because the point of contact falls close to the hand, thus minimizing the torque on it (as compared to a long knife). This increases the efficacy of the pressure-based method of cutting (it requires considerably more pressure than the usual, sliding method)
Second, the convex shape of the cutter means that the force is applied along a small portion of the edge. This is again useful in pressure cutting.
Third, the cutter is faster, especially when performing long straight cuts with multiple orientations, such as one needs to do when dividing a pizza in wedges.
Fourth, it is easy to make sure that one leaves out no parts along the line of cut were the cut has not been performed, or only imperfectly so. And even if one makes the cuts a bit shallow, they will tend to be more uniformly so, and thus allowing one to finish them off securely by pulling a little bit. This would be more difficult to ensure with the knife, as one would need to reset its position several times assuming the pressure-based method of cutting was chosen.
Which brings me to the fifth and most crucial point. The sliding method of cutting is not the most adequate because of the following points:
The pizza must be fixed somehow so that it does not slip when cutting, which is inconvenient due to its thinness. Imagine placing a fork on a certain point to stabilize it. If the cut is long enough, at some point along the cut the fork will be far away from the contact point of the knive and, if the knive is on the downstream side, the pizza portion may tend to wrinkle, especially if it is one of those very thin ones. Furthermore, one avoids the ugly puncture marks left behind by the fork.
Second, the sliding motion may drag some of the looser topings, such as olives or salame slices. This is very hard to avoid when there is a generous amount of molten mozzarella beneath them.