Anti-tank vehicles made their first major appearance in World War II. In German and Soviet designs the traditional turret was removed from an existing tank design, and a larger gun was mounted with a limited traverse in the hull. The weight and space savings of removing the turret allowed a smaller chassis to carry a larger gun. This was used by both forces to keep older designs competitive in the rapidly up-armoring of all AFV's that took place during the war.
For instance, the German Panzer I was obsolete before the war even started, with thin armor and only machine guns for armament. Yet they were forced into battle during Operation Barbarossa, where they were found to be deathtraps. Crews took to using captured Soviet 76.2mm anti-tank guns on them in various makeshift mountings in order to allow them stand off from their opposition. In this form they were later modified at the factory, becoming the Panzerjager I. Soon the same sort of thing happened to the Panzer II, producing the Marder II, and so forth until even the late-war Panthers were being up-gunned in this manner.
United States designs, and British ones based on them, were very different in conception. In pre-WWII planning, US tanks were not to fight enemy tanks directly. Instead the tanks were dedicated solely to the role of infantry support in a fast moving battle. In order to deal with the enemy tanks they would instead rely on tank destroyers, organized into separate groups, dealing with armor at long range.
The resulting US designs retained the turret, but left it open on top for more working room with the larger gun. The larger guns required a weight to be added to the rear of the turret, which can be seen on designs like the M10. The open top made them particularly vunerable to even handguns, and the very idea of independent anti-tank groups was found unworkable. Soon the "basic" Sherman was being upgunned on its own.
Post-war designs have focused almost entirely on weight. With the weight of combat capable tanks growing to the 50 to 70 tonne range, airborne forces were unable to deploy reasonable anti-tank forces. The result was a number of attempts to make a light vehicle with either recoilless or wire-guided missiles as armament, including the Ontos and Sheridan. However neither of these had the armor needed to protect them in a fight, so more modern designs simply place the same weapons on a normal APC chassis. For example, the US Army has vehicles (such as the M113 armored personnel carrier), which may mount anti-tank missiles for defense, but that is not their primary role.