The difference between a mole and a wart is that moles are not contagious, but warts are.
A mole, or nevus, is a collection of skin pigment cells. When they
are small, they look like small brown dots. When the collection is larger, and slightly deeper, it can cause a raised skin colored bump over it. This is called an "intradermal nevus" because it resides in the dermal layer of the skin. These are also called "beauty marks" or "birthmarks" and are not dangerous if they have normal, healthy cells. However, moles can turn bad on occasion, and then they are known as "dysplastic" or possibly "melanoma". Most moles are completely benign and stay that way.
A wart is actually a collection of virally infected cells clumped together in the skin. On the arm or body, a common wart is called "verruca vulgaris" (vulgaris means common). On the soles of the feet, they are called "plantar warts". Warts are caused by the human papilloma virus, HPV, and are mostly harmless. However, certain HPV types can rarely cause malignancy, so any unusual wartlike growth that doesn't respond to regular treatments should be examined by a dermatologist. This is especially true in the case of another HPV-related wart, the genital wart, which can sometimes cause cervical cancer in women.
Moles and warts are both common, usually harmless skin growths, but both should be checked by a dermatologist just to be sure.