From what I can tell NATO tanks usually tend to have bigger turrets with manual loading while Soviet tanks had smaller turrets with autoloaders.
What were the other differences, and why did both sides design their tanks that way?
What were the advantages and disadvantages to these designs?
Although one may argue chicken and egg as to the cause (see discussion between godofthecopybooks and xpyrolegx), the effect is that Soviet and NATO doctrine differed, and their different implementation led to the differences in design. I always caution people against trying to make direct comparisons between NATO and Soviet designs because a Westerner is going to evaluate a Soviet tank by Western criteria. In my reviews of tanks, I perpetually get Russians wondering why on earth I consider long-term comfort, or the ability to ride around with me head out the hatch, to be important. For Soviet purposes, such things are pointless, and they don't get considered. See my opinions on such grading here. http://worldoftanks.com/en/news/21/chieftains-hatch-devils-due/
Bear in mind that the strategic positions of the USSR and NATO were entirely different. There was plenty of room between West Germany and Moscow, not so much between East Germany and Bonn. Soviet doctrine stressed the offense. Even when defending, the purpose of defensive units was to disrupt and attrit attacking units to allow an offensive movement to proceed. That's where the tanks came in, they were the offensive punch. NATO, on the other hand, had no space to trade for attrition. Units would usually have to defend in place a lot more, and the equipment was designed accordingly.
Thus, consider the requirements of a NATO tank, sitting on a ridge waiting for the Soviet juggernaut, possibly for long lengths of time. The tank has to be big enough to be comfortable. It needs plenty of ammo. When defending, a large tank has more gun depression, allowing it to remain more hidden. When retreating, chances are they will be falling back through prepared areas, with bridges and bridging capable of handling heavy tanks, the bridges being blown behind them. Shorter lines of supply allow for higher consumption (and easier delivery) of fuel for large vehicles. And tanks were the primary anti-tank system for most NATO countries.
For the Soviets, the reverse is pretty much true. An assaulting unit is going to have to cross over hastily created bridges, making lighter tanks preferable. In extremis, all Soviet tanks can snorkel and drive underwater across a river. Some NATO tanks could as well, but it wasn't a requirement. A smaller tank with a smaller engine will consume much less fuel, which has to be brought along with the attacking force as the supply lines get longer. (See The Western Desert campaign in WW2). If you're not worried about defending too much, depression isn't a problem, so you can make your tank smaller, and thus harder to hit in the open. Sure, it's a bit more cramped, but if you're attacking, you can time when you get into the tank which reduces the problem a bit. The tank was not considered to be the primary anti-tank system for Soviets, that was the AT gun and the ATGM later.
The auto loader system was started with the assumption that the battlefield would turn nuclear, and crews would reduce in strength. Auto loader didn't have that problem. A second benefit was that with one less crewman, you could make a smaller tank, less volume to protect. Less volume meant that you could have thicker armor per ton of vehicle weight. NATO never really got into the autoloader business prior to LeClerc, barring some oscillating turrets on the AMX13s and the like.
A relative lack of accuracy was the cause of the gun-launched ATGM in T64 and onwards. It was accepted that Soviet tanks would have to deal with things like ATGM vehicles and, if necessary, helicopters, the only way to hit at range was to guide the munition. Western tanks had better systems, making this un-necessary.
Hopefully this all gives you an idea of the different requirements. It goes far beyond simple technical capability, itself an under appreciated subject. The Soviets were very willing to try advanced technological solutions to problems, just not the same ones as the West. Examples, http://worldoftanks.com/en/news/21/chieftains-hatch-devils-due-pt-2/
While someone else might shed light on the specifics of individual tank designs, I can speak on to the different methods that the two groups used when designing their tanks. The NATO countries focused on heavier more technologically advanced tanks, sacrificing numbers for combat effectiveness. The Soviets leveraged their huge industrial capacity to crank out large numbers of tanks. They favored using large armies of smaller lighter tanks to completely overwhelm opposing armies.
Malcolm Chalmers and Lutz Unterseher Argue in their article "Is There a Tank Gap?: Comparing NATO and Warsaw Pact Tank Fleets",
. This NATO advantage is a result of at least three factors: (a) the NATO lead in development and application of key technologies, such as thermal-imaging night vision, fire control computers, high precision guns, and composite armor; (b) the Soviet decision to emphasize the production of larger numbers of smaller and cheaper, but less effective, tank models (the average Warsaw Pact tank weighs only 38 metric tonnes, an average NATO tank, 49 metric tonnes: a difference that is reflected in the combat potential of individual tanks on both sides); (c) the continuing existence in the WTO's fleet of large numbers of older models of tanks-the T-54, T-55, and T-62- most of which have not been significantly upgraded and which are as a result falling further and further behind the standard of NATO tanks of comparable age such as the M-48, M-60, and Leopard 1.
You've got to remember that there NATO had different countries with different reasons for building their tanks the way they did. For example Britain started the cold war with the comet and the centurion simply because that was what it was building at the end of WW2 and various factors meant the attempt to build a post war tank went nowhere. The Centurion was simply what was available.