The term area is generally reserved for a two-dimensional polygon, but surface area is the term of the area of the different surfaces on a three-dimensional figure---the faces on the object. So no, they are not the same.
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figures with the same volume does not have the same surface area.
It is the area of the exposed surface. So if you take a cube, the surface area is the sum of the areas of each of the 6 faces. They are all the same in the case of a cube. In general, add up the surface area of each exposed surface and this is the total surface area.
A sphere has less surface with the same diameter.
Unless the context suggests otherwise, the two are the same. However, you may, for example, be required to find the curved surface area of a cylinder and [then] the total surface area. In that case the total surface area would include the areas of the two end faces.
A surface area of 804.2 cm2