Soviet general strategy developed immediately prior to the war relied on a principle called "Deep Battle". In comparison to German strategy that developed during the first period of the war, this was not about exploiting and pushing weaknesses in particular sectors, but rather, heavily engaging the enemy across all sectors. This limited the effectiveness of breakthroughs in terms of depth penetrated, but also theoretically crippled the ability of the enemy to respond to said breakthroughs by preventing forces from re-orienting to said breakthroughs, making their potential consequences that much more drastic. If it wasn't clear, this relied on substantial quantitative superiority across the front being attacked, as to mount an offensive across a large area meant that particular sectors were not prioritized. In practice, this meant that armoured units were not necessarily concentrated, and instead, distributed piece-meal. Now, in France this type of piece-meal distribution of armoured assets meant that the Germans could quickly overwhelm sectors by local concentration of firepower. However, the sheer number of Soviet armoured units (and the use of a high number of dedicated anti-tank guns and Tank destroyers) meant that, particularly during the later stages of the war, the Germans had great difficulty in developing that local superiority.

In terms of broader strategic aims, the German tactics were direct intellectual descendants of late 19th century Prussian tactical writers who advocated for battles of annihilation, which required the surrounding and destruction of enemy armies. However, Soviet deep battle didn't require that; rather, the broad offensive and breakthroughs were intended to sap the morale and fighting ability of a large unit and force it to withdraw, hopefully in disorder. If I could make a comparison you might be familiar with, it is similar in idea to the 'Hammer and Anvil' strategy of Alexander; the broad operational offensive would hold enemy units in place, while breakthroughs (like Alexander's cavalry charges) would force a collapse of frontal integrity and force a retreat. In comparison, the Germans were constantly aiming for the double envelopment of Cannae in their battles to make up for quantitative inferiority. Unfortunately, this all too often relies on leadership blunders of the opponent. The Soviets provided this in spades early in the war, but later in the war, the strategic mobility of the Germans was significantly reduced. This was reflected in shifting German armoured tactics as the war dragged on. Priority was given over to TD production, which theoretically were able to fire from better cover and was a cheap attempt to bring quantitative parity up to the Soviets. However, they were very ineffective on the offensive, and very vulnerable to any breakthroughs thanks to their lack of a turret.

The greatest weaknesses of the Soviets pre-war were poor equipment (even the flagship and powerful tanks like the KV-1 and T-34/76 were lacking in some serious ways) and training, despite numerical superiority. It would take until 1942/43 for Deep battle to become effectively used.

Now, as for the Americans, they had their own ideas about tank warfare that in many ways were reflections of the British doctrine of Cruiser and Infantry Tanks. If you are not familiar with that doctrine, it involved utilizing the infantry tanks (heavily armoured, slower, and often sporting guns designed primarily to defeat fixed fortifications) would assist infantry in a breakthrough, while cruiser tanks (less heavily armoured, faster, and often with lighter, high velocity guns) would utilize the breakthrough to flood through the flanks of the enemy. In practice, this didn't really work that well. With the advent of man-portable AT weapons as well as quickly escalating armour, the cruiser tanks were found to be vulnerable and underarmed.

Now, the Americans took this idea and turned it on it's head a bit. Instead of infantry tanks they focused on a large number of medium tanks designed to fulfill sort of the role of both the cruiser and the infantry tank. US forces were much more mechanized than their German opponents, so it was thought that speed would be the best offense against heavier German tanks. We can see the dual purpose thinking quite literally in the first American tank used in the war, the M3 Lee: http://upload.wikimedia.org/wikipedia/commons/thumb/8/8e/M3grantmini.jpg/1280px-M3grantmini.jpg The requirements for a tank that fulfilled both the cruiser and infantry role are pretty clear here. On the other hand, the Sherman did away with using two separate guns and instead it utilized a single multi-purpose gun, the 75 mm Gun M2. It was a medium velocity gun gun. However, even by 1944 before Normandy, many realized it was woefully inadequete for dealing with German heavies like the Tiger, and even the Panther. This led the British to retrofit some of their Shermans with the QF17pdr, a 76mm gun with much higher velocity. These higher velocity guns were worse for HE shells, but much better as AT guns, which meant they didn't serve as great a purpose attacking fixed positions, but in tank battles showed their worth. On the other hand, US Army Ordinance was stubborn about reequipping the Shermans (and fielding a heavier tank) and instead felt the best solution was a high speed TD fleet that could quickly be called on to re-deploy and attack enemy tank formations when needed. Except, they were very lightly armoured. This is where I make the connection to the Infantry/Cruiser tank dilemma. The US doctrine of using TDs as tank assassins certainly worked in some cases, but as mentioned, the Shermans (and particularly the TDs) were extremely vulnerable to even man-portable AT weapons. The sort of fast breakthroughs and maneuver warfare that the US doctrine dictated was found to be impractical for most of the campaign on the western front, and indeed, the greatest lesson all sides learned from the war is that a good combo of armour, firepower, and speed in a general tank design was the best option. TD's were fast enough to get into position against panzers to blunt offensives, but too vulnerable to flank or lead them. I would say that the quick effort to upgun Shermans as the war went on, and the quick change to rapidly deploy M26s to Europe was an acknowledgement that the American doctrine pre-invasion was a failure.