In "The Things They Carried," how do the two worlds contrast for Kiowa?

I am assuming you are refering to Kiowa's background as a
Christian but also his Indian heritage. Kiowa is a very interesting character in this
excellent story because of this division within him. He shows himself to be someone who
has a fervent belief in Christianity, but also it is clear that this is not the be all
and end all to his beliefs, as his Indian heritage is also very important to him. Note
how he is introduced with some of the things that he
carried:

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Kiowa, a devout Baptist, carried an illustrated
New Testament that had been presented to him by his father, who taught Sunday school in
Oklahoma City, Oklahoma. As a hedge against bad times, however, Kiowa also carried his
grandmother's distrust of the white man, his grandfather's old hunting hatchet.
Necessity dictated.

Note the
way in which Kiowa is shown to be hedging his bets somewhat, placing faith in both his
Christian upbringing but also his Indian heritage to see him through this war. The
phrase, "As a hedge against bad times," indicates the way in which Kiowa believes that
you might as well place your faith in as many things as possible, as it definitely can't
hurt. The contrast then is between two different sets of beliefs as Kiowa interestingly
chooses items to carry that reflect his own belief in Christianity but also his Indian
roots.