How are Rosh Hashanah and new years day alike?

The similarities: 1) Rosh Hashanah is a new year's day: it is the first day of the Jewish (the Hebrew) calendar year.

2) Many Jews make resolutions on Rosh Hashanah.

3) Rosh Hashanah is a holiday on which Jews don't work.

The differences:

Other than the above, the two are very different. Rosh Hashanah is spent largely in the synagogue. It occurs in September or early October, and is the first two days of the month of Tishrei. Our traditions state that at that time the world is judged for the coming year (Talmud, Rosh Hashanah 16a); and we read the Torah and say prayers which ask for a good year and which declare God's kingship over the world. The shofar (ram's horn) is blown (Leviticus 23:24; Talmud, Rosh Hashanah 33b-34a). After services, festive meals are held in the home, but not parties in the style of New Year's Day.