The question, as stated, is too broad. "Russia and Eastern Europe" is enormous with varied terrain. From the flat, open Ukrainian Steppe, to the rough Caucus Mountains, to urban fighting in places like Stalingrad. "Post D-Day" can be anything from fighting in Normandy hedgerows, to the vast sweeping breakouts of July and August 1944, to the grinding attrition of the Hürtgen Forest, to the rough wintery fighting around the Ardennes.
And both fronts happened at very different times. By the time the Western Front is re-opened in June 1944, fighting had been happening in the East for three years. Tank combat evolved greatly in those three years. Tank combat in the East began with thinly armored and lightly armed Panzer IIs and IIIs encircling of masses of poorly lead Soviet light tanks, occasionally struggling against a lone heavy KV-1 or well deployed T-34. It ended with monstrous German heavy tanks and concealed tank destroyers fighting delaying actions against overwhelming Soviet medium and heavy tanks.
We need a point of comparison in time and space. In time, I'm going to pick June/July 1944. In the West, fighting is mired in the Normandy bocage until their breakout at the end of July.
In the East the Soviets are smashing through the German lines from the Baltic to the Ukraine to Poland in Operation Bagration.
Because Operation Bagration covers so large an area, I will limit myself in space to the stereotypical open steppe country.
These pictures sum it all up.
Normandy bocage: ancient farmland consisting of small open fields...
...separated by sunken roads.
As you can see, one is very hemmed in and one is very open.
Ideal German tactics in the bocage would employ a hidden, turretless tank destroyer, probably a Sturmgeschütz III, or a towed anti-tank gun, firing down the most likely lane of approach. This would be backed by a few crew served machine guns, plus infantry armed with submachine guns (high rate of fire at short range) and hand held anti-tank weapons (probably panzerfaust) to mop up.
The relatively thin armor and weak guns of the Western Allied tanks, plus the bocage forcing them into a frontal attack at close range, allowed the Germans to employ the relatively cheap Sturmgeschütz. The low silhouette of the Sturmgeschütz lent it to ambush tactics, while the hemmed in ground meant the lack of a turret was not a serious disadvantage. German tank crews, fighting in defensive positions without a lot of maneuvering, required less training than the mobile warfare on the Eastern Front.
Infantry could use the bocage to infiltrate around the flanks of the road-bound Allied columns. In an ambush, panzerfausts would be used to destroy the rear vehicles in the column blocking retreat, and the front vehicles blocking advance. The whole column would be raked with flanking fire from solders in the hidden and impenetrable hedgerows.
A variation is what the Allies called "the hedgerow defense" which used the hemmed in nature of a Norman field to negate the Allied advantages in firepower and mobility.
Any infantry advancing would be mowed down by concealed machine guns, and any tanks would be hit in the flank by concealed anti-tank weapons. Conventional assault by fire and movement did not work, there was no cover. This was eventually countered by combined arms.
An Allied tank would fire white phosphorus into the corners to suppress, obscure, or drive off the German machine gun crews and then keep up suppressing fire without exposing the sides of the tank. Mortars and grenades would suppress the surrounding fields to keep the Germans from taking up flanking positions. Finally, engineers would blow a hole in the hedgerow (using pipes shoved in by the tank) to allow the tank to advance for close support.
Source "Busting the Bocage: American Combined Arms Operations in France 6 June--31 July 1944" by Captain Michael D. Doubler
German tactics had to adapt to Allied air dominance and lavish Allied artillery. To be visible from the air meant death by "Jabos" or "Jagd-Bombers", the German word for hunting fighter-bombers. To remain static in a known position meant getting shelled. Even a solid German defensive position eventually had to move under the weight of Allied bombardment.
On the steppes, range was king. The ideal tank engagement on the Eastern Steppes for the Germans was to employ the superior range, accuracy, armor, optics, and training of their tanks and tank crews to pick off the Soviets at long range.
A German 88mm tank gun on a Tiger I or the 7.5 cm KwK 42 L/70 on the Panther could hit and reliably penetrate a T-34 at 3000 m. The considerably shorter 7.5 cm KwK 40 L/48 used on most other German tanks and tank destroyers could penetrate the front armor of a T-34-85 at 1000 yards.
The Soviet 76mm F-34 of the T-34 struggled to penetrate the front armor of a German tank even at 100 yards. The 85mm ZiS-S-53 fared significantly better if the crew could hit the target; Soviet tank crews were often poorly trained and their optics were poor.
Since German tanks were often outnumbered by the Soviets, German infantry would be used to slow down the wave of onrushing Soviet tanks, prevent attempts to flank, and deal with any Soviet infantry that might have gotten close enough to attack the German armor. Infantry would dig anti-tank ditches and set up ambushes with panzerfausts.
Because the tank arms race on the Eastern Front was so intense, the Germans had to use their heaviest and best armed and most expensive vehicles. However, this did not stop them from sending ponderous and overpowered heavy tanks like the Tiger II to the Western Front to break down in transit or be destroyed by Allied air power.
While the Soviets had air superiority, they never achieved quite the air dominance that the West did. Thus, Germans units could operate a bit more freely in the open. What they did have was lavish artillery barrages, mostly useful in the opening hours of a Soviet attack. As a defense, German armor would be deployed some ways behind the immediate front line to avoid being caught in the initial barrage.
While in the West the Germans were on the defensive, in the East, even in the Summer of 1944, the Germans would still look for opportunities to maneuver and counter-attack. Sometimes this would result in local victories, more often this would sap limited German resources.