Aurora is the Latin name of the Goddess of the Dawn, one of the very few gods that have been inherited since Proto-Indo-European times. After migrations, the Indo-European people have adopted most of their gods from other populations, preserving only a few of their ancient gods, like the God of the Sky, Greek Zeus = Latin Jupiter and the Goddess of the Dawn, Greek Eos = Latin Aurora. Despite superficial differences, the Greek names and Latin names of these god and goddess are the same, descending from a common form through phonetic alterations (Dieu => Zeu- = Ju- and Ausos => Eos = Auror-, through regular phonetic changes, while -piter means father and -a was added to Auror to make it more similar with most other female names).

In the Odyssey, there is a lamentation about how the female goddess Eos = Aurora is strongly criticized for taking human lovers whenever she wants, while when the same thing is done by male gods, like Zeus = Jupiter, that is overlooked or even praised. Therefore this criticism about how promiscuity is treated differently by society for females and for males is millennia old.

You have a point that the Odyssey also mentions athletic competitions, not only the Iliad, but the importance and the amount of space dedicated to describing such events is much less than in the Iliad, where an entire chapter is dedicated for this and there are very detailed descriptions of the techniques used by each competitor.

In the Odyssey, the athletic competitions are just a vehicle used to highlight the physical abilities of Odysseus, demonstrating that even if he is older he can still easily outmatch younger competitors.

In the Iliad, the athletic competitions are described much like a radio or TV commentator would describe the events to spectators, focusing more on the description of the actions of the competitors as a show of skill, than on the results.