The most important issue that Western Europe had to deal with during the Cold War was their proximity to the Soviet Union. The arms race between the USSR and the USA would have left Europe feeling vulnerable. Although there would have been some security in having NATO, living next door to a hostile country would have caused a lot of tension.
Of course, the Cold War covers a long time and Western Europe covers a lot of territory, so it's pretty hard to generalize about this. But here are some things you can say:
- For much of the Cold War, Western Europe worried about attack by the Soviets. This was especially true early on.
- Also early on, the area had to expend huge amounts of resources on rebuilding from WWII.
- Later on, these countries had to decide how closely to ally themselves with the US. They had to decide, for example, if they wanted nuclear weapons and/or American bases on their soil.
- They often had political turmoil, especially in France and Italy. Many countries had strong communist parties that allied themselves with the Soviets.
As I say, the problems they faced were different depending on when and where you are talking about, but these were some of the problems.
The Soviet Union, at the conclusion of World War II, had installed pro-soviet governments on its western border. Winston Churchill was the first to use the term "Iron Curtain" to describe the division within Europe. In 1956, Hungary was invaded by the Soviet Union after the students rioted and moved political leaders to withdraw from the Warsaw Pact. The rebellion was quickly crushed, and served as an example to any other bordering country of what the cost of rebellion would be.
The US and allies, having "liberated" Western Europe, continued to maintain democratic governments and had established NATO.
What's interesting to note are the troop and materiel concentrations along the Iron Curtain. The Soviets had several times the amount of conventional weapons (tanks, troops, planes, etc.) than the West. Instead, all the countries on the western side had huge missile installations, armed with both nuclear and conventional warheads. Strategically, this suggests that the Soviets, if given the chance, would move westward; it also suggests that the West would not move eastward, but would risk nuclear war to preserve Western Europe from Communist control. So the region lived in fear of nuclear annihilation for most of the Cold War.