The US and its Cold War allies wanted to use West Germany as a strategic partner against the Soviet Union. The Soviet Union wanted to stripmine East Germany for reparations. Before the end of WW2, the Soviet Union had received large amounts of military aid from the Allies. After WW2 Europe continued to receive aid in the form of the Marshall Plan, which provided heavy discounts on US exports and loans that were mostly forgiven later on.
West Germany's economic boom was in part due to the German government expecting to have to pay the loans back in full and thus setting up a public fund that provided credit to businesses and the rebuilding effort rather than paying them out as direct subisidies to industry. When the loans were eventually forgiven this created a massive windfall. The successor program to the Marshall Plan was explicitly tied to "defeating communism" and focused on military development, which resulted in West Germany rearming despite earlier decisions that the country should be demilitarized. This change of plan was extremely controversial within West Germany and debates around the military involvement of Germany continues to this day.
In East Germany on the other hand, the rebuilding effort happened despite the Soviet Union not with the help of it. Reparations continued even after a lot of the industry had been dismantled and shipped to Eastern Europe while the isolation from Western Europe provided limited opportunities for export and the scarcity of resources meant a lot of early industry was built to provide for the population rather than trade.
Given the conditions, East Germany performed remarkably well, although of course it had a despotic government that would literally shoot its own citizens rather than allow them to leave for greener pastures. The US was also extremely hands-off on West German politics after the immediate occupation era. Presumably a major factor was that the US didn't have as much skin in the game as its European allies whereas e.g. France literally had been occupied by Germans during the war, resulting in much stronger resentment.
It's also worth noting that one of the plans for post-war Germany by the Western allies was full deindustrialization to a greater extent than what the Soviet Union ended up implementing with the explicit goal of turning Germany into an agrarian state incapable of forming a military for the forseeable future. This became less interesting as the divisions between the Soviet Union and the Western Allies became clearer but it wasn't entirely unlikely.
