How was the Truman Doctrine applied differently in Asia compared to Europe? Why was this difference present?
The Truman Doctrine was first applied as a rationale for military and economic aid to Greece immediately after WW II, when communist pressure from Albania threatened the outbreak of another large war. This was also the beginning of the "Domino Theory", originally applied to the possible loss of Greece to communism leading to the loss of Turkey. The US and its Allies could not afford the loss of more countries, particularly one as large and potentially economically important as Turkey. In Europe, the Truman Doctrine also included the Marshall Plan, a system of economic and material aid to rebuild countries destroyed by the War.
In Europe the masses of NATO and Warsaw Pact forces in close proximity required more delicacy in the actions taken involving the Cold War. As Europe was rebuilt and, inevitably, contact between East and West was required, the course of events led to more diplomacy and less force. In Asia, the possibility of escalation to nuclear war was less. Nuclear weapons would not have been applicable in the Korean War, or those in Indo China, so military action (of "regular force" and counter-guerilla war) was used.
The Military Assistance Commands in Greece and elsewhere had stopped overt communist aggression in Europe, and the Iron Curtain had sealed Western and Eastern Europe from one another in most ways. The Truman Doctrine at this point had become largely the theory of "Containment", the idea of resisting the spread of communism wherever it appeared. This was more subtle than it appears, as Containment was not really aimed at military confrontation so much as at spreading thin the economy of the USSR. The theory was that the weight of the World Revolution would sooner or later cause that economy to collapse. The more communism appeared to be winning, the more stressed the Soviet Union's economy became.
This led to economic and cultural overtures to Eastern Europe, and the gradual easing of tensions until the age of Detente arrived in the 1970s. In Asia, Containment continued, culminating in the Soviet-Afghani War of the 1980s. President Carter and the British initially began covert aid to the mujahadeen, and Reagan continued. The military failure of the Soviet Army in that war became the final economic nail in the coffin of the Soviet empire.