What were the primary differences between Soviet Sub designs and US sub designs during the cold war? what "Doctrines" did those designs build towards?

There are two aspects of submarine design that I'll speak to in this answer. The first is the design itself: how the submarines are constructed, how American and Soviet submarines physically differ and how their design philosophies or doctrines differed. The second is the "meta-design": how the design organizations in the different countries had vastly different in how submarines were designed and built.

The Physical Design

The biggest American and Soviet submarines in their design is the way their hulls are constructed. For the most part, American submarines have what is called a single hull. The pressure hull, which contains the crew, reactor, turbines, torpedoes, electronics etc., is the outside surface of the submarine for most of its length (a Permit class submarine shown here; the bold line is the pressure hull). This permits the smallest surface or wetted area for a given displacement (the measure of the volume of a submarine), which decreases drag and increases speed, but the ballast tanks cannot be made very large and there is less room to store equipment outside the pressure hull. The pressure hulls of American submarines usually only had three watertight compartments.

The Soviets used the double hull almost exclusively. A double hull has the pressure hull nested inside a thinner outer hull (Pr. 705 Alfa class shown here). This allows for large ballast tanks to be placed around the pressure hull and equipment to be stored. Double hulls are more expensive to make and maintain and they have more drag for a given displacement, although the outer hull can be nicely streamlined because it does not have to be shaped to bear the pressure of the deep. Soviet pressure hulls usually had at least six watertight compartments.

The Soviets very much valued survivability in combat. Each of their submarines was designed to have one of their compartments and the surrounding ballast tank flooded, be able to get to the surface and stay afloat until rescue arrived. It is notable that all ~~four~~ three Soviet nuclear submarines that sank made it to the surface before they eventually succumbed. Most Soviet submarines also had large escape capsules that could fit the entire crew and most also had automated life raft dispensers. It is still a mystery why so much emphasis was placed on survivability by the Soviets. The rest of the USSR wasn't known for placing soldier safety above all other concerns.

American submarines, by comparison, could easily sink if a large enough hole was put into one of the watertight compartments. The ballast tanks (which themselves are smaller than on Soviet boats) are on the stern and bow instead of distributed, which could potentially make the submarine unstable with heavy flooding. American submarine safety relies upon a superior build quality and excellent crew training. It's worth noting that neither of the two American submarines that sank during the Cold War made it to the surface.

Soviet submarines are generally very streamlined and carefully shaped to minimize drag, noise and non-acoustic signatures. American submarines are usually very blocky and simple looking on the outside because they prioritize simplicity and efficiency of construction. The Soviets emphasized streamlining because they had developed sensors that could detect the turbulent wakes of submarines and they wanted to have submarines that produced the smallest possible wake if the Americans developed the same technology (they still have not, as far as I am aware).

Because of the very poor quality of the ~~two~~ three-year conscripts that made up the bulk of the enlisted men, the Soviets placed a heavy emphasis on automation. The Alfa class attack submarine had a crew of only about 30 due to the extensive use of automation. American submarines, in contrast, are large designed to always have a man in the loop. This is why a modern Russian submarine has a crew of about 60 and an American Virginia class submarine has a crew of about 130.

The Soviets were not afraid to try new things. They cooled the propeller shaft seals on the Akula class to minimize rubbing noise. They used active noise cancelling to make their submarines quieter. They experimented with titanium construction. They made a giant submarine with two parallel hulls and a displacement greater than the Bismark (the Typhoon SSBN). The list goes on and on. The US stuck to a proven design that worked, but might not have offered the most cutting edge technology (besides sonar and acoustic quieting where the US had a distinct edge up until the 1980s). The Soviet would produce multiple classes at the same time, which was great for innovation, but very expensive. The US only produced one attack submarine and one ballistic missile submarine class at a time.

In general, Soviet submarines can be distinguished by their innovation whereas American submarines tend to be of a much more conservative design, which leads me on to the next topic.

The "Meta-Design"

In the United States, submarines were essentially designed by the Navy's own naval architects (BuShips and later NAVSEA). The shipyards had some input too. The driving force behind nuclear submarines was Admiral Hyman Rickover, a controversial figure to say the least. Rickover was ostensibly only in charge of the reactor design, but in practice, he was the dominating force over American submarine design. Rickover was obsessed with safety and the conservative designs of American submarines reflect that. The atmosphere was military, but even more hierarchical and constricted than usual because of Rickover's domineering presence.

The Soviets, on the other hand had a much more interesting system. They had several design bureaus, similar to the aircraft design bureaus like MiG, Sukhoi, Tupolev in the USSR. The closest American equivalent would be companies like Boeing, Lockheed Martin and Northrop Grumman. Although each design bureau had a speciality (Malakhit designed attack submarines, Rubin designed missile submarines) they were fiercely competitive. This lead to very interesting and innovative designs. In addition, the Soviets employed many more designers than the US and out-of-the-box ideas were encouraged. The atmosphere was much academic than military (in fact the head designers were often members of the Soviet Academy of Sciences).

A good summation of the US and Soviet systems was articulated by a Soviet designer: "We had competition in submarine design. You [with Rickover] had Stalinism!"

TL;DR: The Soviets emphasized survivability, automation, streamlining and general innovation. Their submarines were designed by several competing academic teams, leading to interesting and innovative design. The Americans emphasized simplicity, reliability and safety. Their submarines were designed by a central, military controlled design bureau.

Sources

American Submarines Since 1945: An Illustrated Design History by Norman Friedman

Cold War Submarines: The Design and Construction of U.S. and Soviet Submarines, 1945-2001 by Norman Polmar and K.J. Moore

Fire at Sea: The Tragedy of the Soviet Submarine Komsomolets by D. A. Romanov

Submarines of the Russian and Soviet Navies by Norman Polmar and Jurrien Noot

Rickover: Genius and Controversy by Thomas B. Allen and Norman Polmar