I would say that the main difference between Woody and Mr. Wakatsuki is in their attitudes towards the United States. Mr. Wakatsuki is not a citizen (is not allowed to be a citizen) and therefore feels really rejected by the country. By contrast, Woody feels that he has opportunities in the United States and that he must pay the country back for those opportunities.
You can see this difference best in the argument that the two have over whether Woody should volunteer to go in the Army. Woody thinks it is his duty. Papa is still angry at the US for the way it has treated him and does not want Woody to volunteer.