You definitely do not need to use a tarp. You do not need to use it as a rain fly and you do not need to use it underneath the tent. However, there is some benefit to using the tarp (even if it is raining). If you are camping on a very sandy/dirty surface you will want to use the tarp to keep your tent cleaner than mud. The tent is completely water-proof so no need to put it over the tent. In extended use (months at a time-not recommended by the manufacturer) it can help prolong the life of the tent.
If you are camping on grass there is usually no need to use the tarp.