These two major theories of international relations would give very different explanations for the start of the Cold War. This is because they explain the behavior of states in very different ways.
Realism focuses only on the security of the state. It argues that states base their actions on the desire to maintain their power and security in the international arena. They act solely in ways that are meant to accomplish this. Therefore, a realist would argue that Russia, if it has any serious degree of power and ambition, will always want to control its surrounding areas. It will feel that controlling these areas will protect it from attack and give it a base of power. For this reason, post-WWII Soviet leaders wanted to control Eastern Europe. Their actions threatened the United States and made it feel less secure. Therefore, it became determined to stop communist expansion and the Cold War ensued.
Constructivism focuses on the way in which different states conceive of themselves and the rest of the world. Realism says that the type of government a country has, or its attitudes towards other countries, does not matter. All that matters is security and power. Constructivists say that governments and attitudes do matter. They say that countries come into conflict because of the ways in which they define themselves. In this view, the Cold War came about because each side defined the other as the enemy. Each defined the other as a threat to its own side’s very existence. Each defined the other as less moral and in some way evil. These attitudes led to aggressive policies that brought on the Cold War.
Thus, these two theories have very different explanations for the Cold War. One says it was all about power while the other says it was brought on by perceptions and attitudes.