The Old Testament, the holy book shared by the three Abrahamic religions, Judaism, Islam, and Christianity, was originally written in Hebrew. Christianity diverged from its Jewish roots by accepting Jesus Christ as the Messiah predicted in the Old Testament, and added a second body of Scriptures known as the New Testament. The books of the New Testament were written in Greek, but many reflect an Aramaic substrate. Islam, while accepting both the Old Testament and acknowledging Jesus Christ as a prophet, added a third scriptural text, the Koran, composed in classical Arabic. All of these three terms, "Holy Spirit," "Holy Ghost," "Spirit of God," translate various conceptions into English, with different meanings depending on the particular religious tradition.
In Judaism, the term "holy spirit" or spirit of God refers to the effect of God or the way God inspires or enfuses the world. It does not refer to a separate person. It is rarely used in the Koran but angels can sometimes be called spirits of God, although "messengers" (the literal meaning of the Greek word ἄγγελος) is more common.
The terms "Holy Ghost" and "Holy Spirit" are used by Christians to refer to the third person of the Trinity (Father, Son, and Holy Ghost). Older translations of the New Testament, such as the Authorized or King James version, use "Holy Ghost" and more recent translations use "Holy Spirit."
Essentially, this stems from the English translators from the middle ages. They created "Holy Spirit" and "Holy Ghost" to reflect their belief that they are different entities in the English bible.
The "Holy Ghost" represents the third person in the Holy Trinity, whereas "Holy Spirit" refers to the Spirit of God or Spirit of the Lord encountered by the Hebrews and Jews in the Old Testament.
So, essentially, "Holy Spirit" and "Spirit of God" refer to the same thing.
Interestingly, these translation and printing differences do not exist in the Bible itself, in either Hebrew or Greek.
Most modern translations have discontinued the distinction. Now, "Holy Ghost" and "Holy Spirit" essentially mean the same thing.
Source: years of theology class in a Catholic School :)