So, are they the exact same in qualities in both Roman and Greek mythology?
If not, what qualities did they share?
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Yes Apollo is the same god in both Greek and Roman mythology.
From Wikipedia The Roman worship of Apollo was adopted from the Greeks. As a quintessentially Greek god, Apollo had no direct Roman equivalent, although later Roman poets often referred to him as Phoebus.
A similar question was asked here: Was Apollo the sun god in Augustan Rome?
The Roman story was that they adopted the god Apollo from the Greeks after a plague in the 430s BCE. According to Wikipedia:
On the occasion of a pestilence in the 430s BCE, Apollo's first temple at Rome was established in the Flaminian fields, replacing an older cult site there known as the "Apollinare". During the Second Punic War in 212 BCE, the Ludi Apollinares ("Apollonian Games") were instituted in his honor, on the instructions of a prophecy attributed to one Marcius. In the time of Augustus, who considered himself under the special protection of Apollo and was even said to be his son, his worship developed and he became one of the chief gods of Rome.
The Roman Emperor Augustus had a special devotion to Apollo as well, considering him a patron god. They seem to have seen Apollo as essentially the same god as the Greek version: oracular, artisitic, athletic. However, he took on a more imperial role during Augustus' time.