What differs about the extra day added to Rosh Hashanah is that it is observed in Israel (whereas the extra day of the other holidays is not observed). Truth is, the two days of Rosh Hashanah are not really even seen as two separate days, but rather as "one long day" (yoma arichta in the Aramaic of the Talmud). It is because of this that there is question as to whether Jews should recite the Shehecheyanu blessing on the second night of Rosh Hashanah. Thus the custom of having a new fruit (one that hasn't been eaten yet this season) on the table when lighting the candles and reciting Kiddush on the second night of the holiday. The new fruit gives us a reason to make a Shehecheyanu blessing.
I've always liked this custom since eating new fruits is both delicious and adventurous. There's also no shortage of exotic fruits, especially with new fruit breeding taking place as in the case of Apriums and Pluots.
Last year I posted something about William (the Jewish Robot) Levin's viral marketing animations called "The Adventures of Todd and God". Well, another episode of Todd and God has just been released and it focuses on the custom of eating a new fruit on the second night of Rosh Hashanah!
Here's the new Todd and God video: