Someone I know argued that the use of armor (both German and Allied) in North Africa was closer to conditions on the Eastern Front than it was to the Western Front. German armor tended to be placed in stationary, defensive positions on the Western Front or at least more so than in North Africa, but I'm having trouble seeing a strong connection between North Africa and the Eastern Front (disclaimer: I have a mediocre at best understanding of armored warfare on the Eastern Front). I'd welcome any opinions for or against the claim. Thanks for reading and/or responding.
Both were typically very open battlefields, with no real front lines and huge capacity for maneuver warfare. Infantry played a secondary role to armour because of their lack of mobility.
The scale of fighting on the two fronts was very different, and there were differences associated with, for example, the better communications and co-ordination of the British than the Soviets, and the more 'gentlemanly' atmosphere of the desert fighting.
In both theatres, the style of battle involved armoured thrusts, with each side almost playing a game of nerves: both sides would have forces well behind the 'front lines', and the first army to pull back would lose. But to remain forward risked being cut off from supply and being forced to surrender. These situations also occurred on the Eastern Front.
In Europe, Allied air power forced the Germans to camouflage and limited their mobility. The Normandy terrain was the opposite of the desert or Russia steppe (open, rolling grasslands), favouring ambush and making control difficult. The sweeping strikes were impossible until the Allies broke out of Normandy, but then the Germans were forced back to their own borders, the next defensible positions (and like Russia and the desert, the Allies out-ran their supply lines). After that, it was back to set-piece battles that almost resembled those of 1918. It should also be remembered that 'the Eastern Front' meant all sorts of different situations, endless ones in fact. I have here discussed the stereotypical grassland battles of 1941-late 1943. After that the fighting was frequently in swampy, forested or urban areas. The Eastern Front really had everything, such was its scale.