For much of the American and European (Christian) world, the halls are decked, and 'tis the season to be jolly. Twinkly lights, cheesy spangles, and rampant commercialism fills the air. Possibly in an effort to appear more accepting of alternate traditions, advertisements hocking gifts say "Happy Holidays" instead of "Merry Christmas", or if they say that dreaded C-word they include Chanukah, and Kwanzaa. I've heard many Jews express the opinion that Chanukah is a minor holiday and all the hooplah is purely a figment of the media's imagination.
I think that Chanukah is underrated in terms of the value Jews place on it. Yes, it's a light "party" holiday with a couple very different legends behind it, such that it's easy to be confused as to what, exactly, you're celebrating. But as my Rabbi pointed out, Chanukah is one of the few religious holidays that has historical evidence supporting the events it commemorates: the Maccabee rebellion happened and they did rededicate the Temple after the Greeks desecrated it. My Rabbi also took it one step further: if it weren't for the Maccabees and their reunification of the Jewish people and rededication of the Temple, world history would be completely different.
For example, it's very unlikely that the Christmas story would have even been possible. Due to the fighting at that time between Syria and Egypt, as well the infighting amongst the Jews who supported Syria versus those who backed up the Egyptians, it would have been difficult for the Jewish community to hold a census some 300 years later in the City of David so Jesus could be born there as the prophesies required. It is more likely that the Joseph and Mary would have assimilated and acted like Hellenists instead of Jews, or at the least, been too involved in war for such an undertaking.
I also think that having a Festival of Lights at this time of year is vital, because I live in a place that doesn't have much daylight now that we're approaching Solstice and it's easy to feel a little down because of it. Really, this is one of the reasons why many religions that originated in the Northern Hemisphere have celebrations in mid to late December. It is believed that Chanukah incorporated aspects of a more ancient Solstice practice, but it is known that European pagans celebrated Yule and Christianity chose to celebrate Christmas at a time to coincide with those celebrations in order to suppress paganism (and/or because they felt they really needed a pick me up as the days got shorter).
From a more religious perspective, Chanukah is very much a holiday that celebrates a value near and dear to most Americans: the right and ability to practice religion in a manner of the individual's choosing. When we light our menorahs, we put them in windows in order to publicize the miracle of the oil. We couldn't do that in times of oppression and we're lighting the candles in order to celebrate and commemorate our victory over an oppressive military regime. The Greeks wanted us to assimilate. We wanted our Temple back, so we fought and won. Supporting the idea that it's not just about Jewish freedom is the fact that our victory allowed Christianity as we know it to exist.
Honing in on the Jewish angle, and touching on what you alluded to, Chanukah is a holiday that celebrates the Jewish identity. We came together. We rebelled. We won. We regained the Temple and we experienced the miracles. This is why we bristle at comparisons to Christmas, which in my opinion, is why we tend to downplay its importance. It is certainly the reason that, despite my interfaith celebrations, I reject the term "Christmukah".
Granted, not all Jews celebrate Chanukah, notably those with roots from Iraq and the Eastern parts of the Greek empire. This definitely supports the idea that Chanukah is a secondary holiday. I would never argue that it's as important as Yom Kippur, but I do believe that it has value.