Vow and oath are both quite formal, so we rarely use them in speech. You can say something like, "I made a vow to myself that I would not break my New Years' Resolutions this year!" In this case, it means you are very serious about your intention. If instead you said, "I promised myself . . . " it wouldn't be as serious. You can say, "I took an oath and I will not break it," but again, this means you are very serious about this oath. Using oath instead of promise in this case would mean that are absolutely determined not to break your oath. In nearly all cases, promise would be used instead of vow or oath.

To add to Kate's answer... they are not limited to a religious context, but they ARE limited to formal, serious commitments. In a typical U.S. wedding modeled on Church of England traditions, the promises the couple makes to each other "to cherish and protect, for richer, for poorer, for better, for worse, forsaking all others, till death do us part..." are referred to as "the marriage vows." A British patriotic song is entitled "I Vow to Thee, My Country." When a U.S. president is elected and promises to defend the Constitution, he "takes the oath of office." Nobody would ever use it in everyday context. I would never tell my boss "I vow you'll get my report by the end of the day tomorrow."