Why does the ancient menorah have only seven branches and the modern day menorah have eight branches?

The Temple's menorah and the Hanukkiah (Hanukkah-menorah) are two different things that have a similar appearance.

The Temple's menorah was the seven-branched candelabra (in Exodus ch.25) that used to exist in the Great Temple in Jerusalem. All of the branches of the menorah were of equal height. The menorah itself was massive (eighteen hand-breaths high - roughly five feet tall), made of gold; and it had seven oil lamps that would be lit late every afternoon. Since the Destruction of the Second Temple, there has not been a menorah.

A Hanukkah-menorah (or hanukiah) on the other hand, is the nine-branched candelabra that Jews use to celebrate Hanukkah all over the world. Eight of the branches are of equal height while the ninth is usually higher (it can be lower, but the height must be distinguishable). Hanukiot can be oil lamps, or hold candles, or (where there are safety-issues) it can be electric. They are used during Hanukkah only and the number of candles lit each day corresponds to the number of days elapsed since the beginning of the eight-day holiday. The eight main branches symbolize the eight days of Hanukkah and the ninth branch holds the Shamash or "leader candle" which is used to light the others. The choice of using the eight-flame hanukiah to symbolize Hanukkah is based upon the miracle of Hanukkah, in which a one-day supply of oil for the menorah was miraculously able to burn for eight days (Talmud, Shabbat 21b).

(Note that many Orthodox Jews refer to the hanukkiah as a menorah; and there are halakha-sources which do the same since the Word "hanukkiah" is a recent Israeli invention. However, it may be useful to refer to it as a hanukkiah in order to clarify the difference between the two objects.)