The M4 Sherman was in a totally different class than the Tiger I - for one, the Tiger weighed nearly twice as much. The Tiger I, in a one on one scenario without battlefield factors taken into account, was obviously a deadlier vehicle than the M4 Sherman, featuring thicker armor (if using an obsolete armor scheme) and a bigger, more penetrating gun. However, that's a stupid way to evaluate the relative effectiveness of a tank.
Tanks are merely a component of a quality army - they fill a certain role, but other weapons systems, and the battlefield itself, have an impact on the effectiveness of any tank. The Sherman had several significant advantages over the Tiger:
- It was less expensive, and could thus be built in greater numbers;
- It was more reliable, and thus was more likely to actually be on the battlefield when needed;
- It was more advanced, featuring superior fire control and a more effective armor scheme;
- It was more upgradable - there's a reason no one used a vehicle similar to the Tiger post-war.
The Sherman was more than capable of doing the job it was designed to do - it could kill the majority of its opponents without serious difficulty (most German vehicles by 1944 were the Panzer IV and StuG III), was an excellent infantry support vehicle, and was far more likely to actually be available to fight than any Tiger. A gun with superior anti-tank performance would have been nice, but 3/4 of the shots fired by Shermans were against unarmored targets or fortifications, and the superior high-explosive performance of the Sherman to the Tiger or Panther lent itself well to the type of war the Allies were fighting.
The M4 Sherman was only inferior to the Tiger or Panther if you place it in an arena without other combat systems or maintenance and logistics problems. The M4 was a superior overall weapon of war, much like the Soviet T-34 - the Tiger or Panther might win a single tank duel, but the M4 and T-34 won battles, campaigns, and wars.
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WWII Shermans typically came with one of four guns: the 75 mm Gun M3, 76 mm Gun M1, the QF 17 pounder (just over 76 mm), and the 105 mm Howitzer M4.
The 75 mm Gun M3 was mounted on earlier versions of the Sherman. It fired several shells of 75 mm caliber - the primary round was the M48 high explosive round (used against lightly-armored targets, like entrenched infantry, anti-tank guns or artillery, or fortified buildings), but the M3 gun also fired armor piercing shells (anti-tank), canister shells (basically a giant shotgun shell for anti-infantry fire), and smoke or white phosphorus shells (for laying smoke or starting fires).
The 76 mm Gun M1 had superior armor-piercing performance, and was carried on many later variants of the Sherman. However, it had inferior high explosive performance (due to the higher velocity the gun fired its shells at, requiring a tougher shell that could carry less explosive), and thus was not always popular. In addition to the HE, smoke, canister, WP, and standard AP shells, the M1 gun also had access to the high velocity armor piercing (HVAP) shell, but those were typically assigned to tank destroyer battalions and were also quite rare, and Sherman tankers rarely had access to those.
The QF 17 pounder was a British gun, mounted in about 2,000 Shermans (nicknamed the "Firefly") given to the United Kingdom. This gun was similar to the 76 mm Gun M1, but its AP rounds carried a larger explosive charge and thus fired at an even higher velocity. This gave the Firefly even better anti-tank performance, but correspondingly made the Firefly even less capable of fighting enemy infantry - the Firefly thus had a more specialized anti-tank role - replacing all 75 mm-armed Shermans with Firefly's (which is a suggestion I've seriously heard made by laymen) would have made the U.S. Army less capable overall, even if it somewhat improved anti-tank performance. Firefly rounds were also physically bigger, and thus Firefly's could carry less ammo (without doing away with one crew member).
The 105 mm Howitzer M4 was meant as a support gun - basically, it was artillery mounted on a Sherman. This gave American armored divisions extremely reliable and integral mobile artillery support. This gun was used to do everything artillery could do - suppress enemy troops, hit fortified positions, and so on - it could also pop an enemy tank like a balloon if one of those shells hit (due to the extremely large high explosive charge carried), but these tanks were rarely engaged in direct anti-tank combat.
Post-war variants of the Sherman managed to mount 90 mm and even 105 mm anti-tank guns (rather than the shorter howitzers).
Note that any Sherman could take out a Tiger I from normal combat ranges - the Tiger's armor really wasn't that good (it was thick, but also flat). The Tiger could engage from a greater distance, theoretically, but not by a significant amount - tank fire control wasn't good enough to reliably hit a target at more than 800 meters.
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The American M1 Abrams and the German Leopard 2 are actually very similar vehicles, with similar combat performance - they generally carry the same guns, similar electronics, similar armor, and so on. The main difference is that the Abrams uses a gas turbine engine, while the Leopard uses a more conventional diesel engine - the Leopard is also faster, due to a different design philosophy which emphasizes mobility a bit more than armor. The Abrams is also somewhat more expensive to maintain and operate. However, the two vehicles are again very similar.