Breakthrough role; and one can debatabley assert that it was a natural evolution of the infantry tank doctrine. The entire point of tanks meant to force a breakthrough or work in close concert with infantry is being able to withstand the naturally heavy fire they would draw from moving into the 'teeth' of an enemy defense. They needn't move very far or very fast, but rather be able to keep pace with the infantry both (a) in speed and (b) tactically, by not being forced to seek cover by the initial shock of defensive anti-tank fire.

In reality heavy tanks were often used for much more than their intended role, but attempting to rely on their heavy armor alone is insufficient - especially if they had to contend with local reserves of enemy armor or dug-in enemy armor. They often tended to have standard-caliber firearms as their role further evolved from that of the Infantry tank: the Matilda, for example, was inadequately armed for mobile warfare with a 45mm, but performed splendidly in close-order attack, yet the IS-2 sported a 122mm cannon, the Tiger a 88mm; striving to contend both with front-line defenses as well as potential reacting enemy armor.

Medium tanks were often equally capable of operating in the combined roles of breakthrough and exploitation, but the natural results were significant attrition for their lack of dedicated role. As I've stated before even the most robust medium tanks, such as all the end-war models used by all belligerents, were prone to dizzying amounts of losses when attacking into the teeth of properly prepared defenses. Comparatively the use of dedicated tanks for this purpose resulted in better performance but the creation of a resource-intensive model of tank (e.g: The Tiger series) with a limited tactical role. The shifting technologies created by total war forced tanks meant for inherently offensive missions to re-define themselves; with the Tiger series of tanks often being used as heavy-hitting "firemen" rushing from one embattled sector of the front to the other - a role which they were admittedly splendid in so long as the prosecution of the battle remained restricted.

Heavy tanks were desirable for their ability to absorb punishment and create room for maneuver, for more flexible tanks, but lacked the accessibility and flexibility of their medium partners. We begin to see a slow shift towards "Main Battle" units that attempt to combine the best elements of both - early attempts such as the T-55, M26/48, Centurion, updated Israeli M-50s, etc. The British Chieftain, the American M60 and the Soviet T-64 being the most visible and immediate results of this quest to combine the best elements of Medium armor with Heavy armor.