As an unrehabilitated American, I grew up viewing Thanksgiving - the one celebrated on the last Thursday in November, not the Canadian one - as a holiday unlike traditional Christian and pagan celebrations. In our shul community, no one would have suggested that a Jew celebrate Halloween, much less Easter or December 25th. Thanksgiving, though, was considered a more pareve celebration in the U.S., and many Jews used the opportunity to have family gatherings, eat turkey and watch football. [For a discussion of the halachic propriety of celebrating Thanksgiving, see the article here; naturally, there are multiple views.]
This is why many American Jews are now making a big deal out of the extremely unusual overlap of Thanksgiving and Chanukah this year; Aish haTorah, for example, offers a full feature on Thanksgivukkah, accompanied by this comedy routine by Stephen Colbert. To many American Jews, Thanksgiving is benign, and therefore the overlap is more notable than, say, that of Pesach and Easter.
Taking this overlap a step further, some have contended that the two holidays are basically similar; both express gratitude to G-d for rescuing us from harm. However, in my view, these celebrations are not at all comparable. American Thanksgiving is an "apple pie" sort of day, on which the average citizen might contemplate his bounty while reclining in an easy chair, one hand on his belly and the other holding a remote control. The theme of gratitude for a good harvest lends itself to a זמן שמחתנו of sorts, a joy at surviving drought and blight with good for the coming winter.
Chanukah, though, is a more grim and complicated holiday for me. The themes of Chanukah are war against brutal foes, assimilation into another culture, Jew battling against Jew for the future of Judaism, the hard process of rebuilding a nation that has been enslaved to another, and the difficult transition of the Chashmonaim from priests to warriors to kings. Thanksgiving smiles with a full stomach (leaving the concern for what colonists did to Native Americans for another day, of course); Chanukah thanks G-d while wiping away blood and bandaging wounds. Thanksgiving is user-friendly; Chanukah is brutal.
Certainly, we celebrate Chanukah with joy and gratitude to G-d, but let's not sand down the rough edges in our desire to have a happy day. Better to think through the challenges posed by the stories of the Maccabim and the Hellenizing Mityavnim, the history of our relationship with the Greeks, and the rise and fall of the Chashmonai [Hasmonean] kings, and emerge from this Chanukah with a better understanding of what those stories and that history mean for our own lives.