Thanks to the many persons who responded to my question regarding these two
terms. My original question was in the context of Ernest Bloch's SACRED
SERVICE where the score has the term "Adonai" throughout, but a recording I
have uses "Adoshem" instead.

To explain the difference I will quote from Steve Barnett of Barr Music
Productions, whose answer seems to sum up what most of my other replies
stated very eloquently.

"Adonai" is how the Hebrew abbreviation for the name of God (either
yod-hay-vav-hay [the famous "tetragrammaton"], or two yods) is pronounced
when encountered in the Hebrew Bible. Since we don't know how to pronounce
the "real" name of God (only the High Priest in the Great Temple in
Jerusalem knew that pronunciation, and it was only uttered once a year in
the Holy of Holies on Yom Kippur--the pronunciation was lost when the Temple
was destroyed by Rome in 60 C.E. and the High Priests were killed or
scattered), "Adonai" is the closest thing we have to the name of God. Since
the name of God should only be uttered in prayer, and (arguably) the public
performance or recording of a Hebrew religious work does not constitute
prayer, a substitute sort-of portmanteau word is used. It takes the first
two syllables of "Adonai" and adds the Hebrew word for "name"="shem" at the
end. Hence, properly in order not to take the name of the Lord in vain, in
performance (or recording), anytime the Hebrew word "Adonai" is used in the
text, the less "prayerful" "Adoshem" should be substituted. However, if you
were performing the Bloch as part of the Friday evening service in a
Synagogue, "Adonai" would be correct and proper.

In addition Benin a box pointed out that this is also true for the term
"Eloheinu" which is changed to "EloKeinu".

Several persons mentioned that since Bloch wrote the piece for a Reformed
Jewish service (which is the only denomination that would use musical
instruments in a service context) that choice of using "Adonai" or "Adoshem"
becomes only a question of whether one wishes to avoid offending the
sensitivities of any Orthodox Jews who might attend the performance.

Terry Sanford
trsanfor(a)home.com