Majority opinion
a.    Only partly.  The creche was a violation of the Establishment
Clause, but the menorah was not.
b.    The Court applied the famous Lemon test to this case.  Under
the Lemon test, a statute or practice which touches upon religion, if
it is to be permissible under the Establishment Clause must:
(1)    Have a secular (non-religious) purpose;
(2)    Neither advance nor inhibit religion in its principal or
primary effect;  AND
(3)    Not foster an excessive entanglement with religion.
The Court reached two different results under this test as to the two
displays.
c.    As to the creche, the Court held it  failed the Lemon test
and thus violated the Establishment Clause.  Lynch held that a creche
display turns on its setting, when applying the Lemon test to it.
The Pittsburgh creche is distinguishable from the creche in Lynch for
a number of reasons.  In Lynch, there was a holiday display of a
creche.  The creche was put up by the city.  The display comprised a
series of figures and objects, each group of which had its own focal
point.  Santa’s house and his reindeer were objects of attention
separate from the creche, and had their specific visual story to
tell.  The “talking” wishing well (“Hello, throw a penny in me”) was
a center of attention separate from the creche.  Finally, the creche
was put up in a public park.  In sum, this display celebrated
Christmas in some manner and form, but not in a way that endorsed the
Christian religion because there was significant diffusion of any
religious message.
The creche in Pittsburgh, on the other hand, was owned and put up by
a private Catholic group.  In addition to the usual figures of Jesus,
Joseph, shepherds, wise men, etc., the creche contained a banner
proclaiming, “Gloria in Excelsis Deo!” (“Glory to God the Highest”).
The creche stood alone; there were no other nearby significant
symbols, such as Santa Claus, reindeer, or “Seasong Greetings”
banners, to diffuse the religious message of the creche. Finally, the
creche occupied a prominent position on the Grand Staircase of the
Courthouse (a prominent building, not just a public park).  This
creche fails establishes religion, fails the Lemon test, and thus
violates the Establishment Clause.
d.    As to the menorah, the Court held it did not fail the Lemon
test and thus did not violate the Establishment Clause.  It is true
that the menorah “is a religious symbol—it serves to commemorate the
miracle of the oil as described in the Talmud—but the menorah’s
message is not exclusively religion.”  Unlike a creche, a menorah has
both religious and secular dimensions.  The Court reasoned that the
18 foot-high menorah was near a 45-foot Christmas tree, the tree was
primarily secular, and the tree was much bigger than the menorah.
The menorah was also next to a “liberty” sign.   Finally, the menorah
was outside of a city building, not inside of a prominent building
such as a Courthouse.  All of these symbols taken together as part of
a single display outside therefore conveyed the message of
celebrating the secular Christmas holiday season, not the message of
celebrating the religious significance of Hanukkah.  That is, these
symbols taken together diffused any endorsement of a religion.  They
do not fail the Lemon test and thus are an Establishment Clause
violation.