To what extent did the reasons for American and Soviet intervention in the Middle East between 1956 and 1982 differ?
The reasons for the superpowers' interventions were fundamentally similar. Both sides wanted to expand their influence in a region of the world that was both strategically important (particularly because of oil) and "up for grabs" in terms of its loyalty to one side or the other in the Cold War.
The only major difference between the two sides had to do with the US's interest in Israel. The US's dedication to protecting Israel had to do with considerations other than simple power politics. American desire to prevent another Holocaust, along with the importance of the Jewish vote in the United States, ensured that the US would be a staunch ally of Israel.
Thus, the two superpowers were both involved in the Middle East for reasons of power, but the US also had other reasons to stay involved in the region.