You are right that the US Constitution does not use the phrase "separation of church and state." I think that this is often used as a short-hand for referring to the First Amendment (I use is that way), but the framework created by the constitution is more complex than that expressed by the phrase “separation of church and state.”

There are two clauses in the First Amendment, which deal with religion: 1) The free exercise clause and 2) the establishment clause. The free exercise clause prevents the government from interfering with or prohibiting religious beliefs and practices. The second clause prevents the government from establishing a religion. Most obviously, this prohibits the government from creating a state church, which had been the practice of European countries. However, the extent to which this prohibition applies to other acts of government, and how its conflict with the free exercise clause, are always sources of litigation. The resulting case law has drawn lines which are not always clear. For example, context seems to matter. There is not a prohibition against public officials stating their own religious views or calling upon God from assistance. Congress can start each day with a prayer and Courts can ask God to “save this honorable court” at the beginning of each session. However, if the context directs religious speech specifically to children, such as a teacher to a student, this is prohibited by the establishment clause. It is argued that the relationship tends to coercive, and prone to making a student feel compelled to accept certain religious beliefs. Thus, prayers initiated by school employees are prohibited. However, prayers initiated by students are generally protected by the "free exercise clause."  A teacher stopping a group of students from holding a prayer circle before school, under most circumstances would be a violation of the students’ rights.

There is also the conflict between atheists and believers. One side argues that atheism is the neutral stand, and thus government and government officials should be prohibited in making any kind of statement or act in any way, which demonstrates or supports religious belief. The counter argument is the atheism is not the neutral stand, but is itself simply another "religious belief system" (like being Catholic). Thus, the promotion of atheistic positions is itself a violation of the establishment clause. There is some Supreme Court case law in support of this position that atheism is itself just one of many religious belief systems (you do not have to believe in God to have a religious belief).

Over time the exact lines in these debates have moved back and forth. The acts and statements of the founding fathers support the proposition that in enacting the first amendment they did not mean to prohibit all public proclamations referring to God or asking for God's blessings. From the beginning, George Washington often made references to God. But, I do think that it is clear that they did not want a system in which God would be invoked as the basis for governmental authority, as had been the case in Europe.

If you look at the language in the Declaration of Independence, you see how Thomas Jefferson (with editing by Benjamin Franklin) turned the old European system upside down.

We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.--That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, --

Rather than asserting that God gave authority to kings, Jefferson asserted that God gave unalienable rights to the individual. The invocation of God in the Declaration of Independence acts as a protector of individual rights. If the rights of an individual were bestowed by the state, those rights can always be taken away by the state. However, if those rights are divine in origin, they can never be taken away. Thus, a public invocation of God in accordance with the Declaration of Independence becomes a protector of rights and not a threat to the rights of the individual.  

This is the argument behind the 1954 congressional act, which established the inclusion of "In God We Trust" on American money. This act was passed at the height of the cold war and Congress wanted to make of statement concerning how the United States fundamentally differed from the Soviet Union. The USSR officially adopted atheism as its religious stance and argued that rights come from the state. Thus, under the Soviet model the state had the right to deny basic individual rights (free speech, religion, etc.) in order to promote to nation as a whole. Looking back to the language of the Declaration of Independence, in 1954 Congress wanted to make it clear that the US was fundamentally different from the USSR, because Americans see their rights as unalienable rights granted by God.  

I think that there is plenty of room to debate concerning specifically where the lines established by the First Amendment should be drawn, or whether at any given time they drift too far in one direction or the other. But, I wanted to give you some of the debate’s historical background and the ideas, which underlie this debate.