the first one means a square and the other one means a cube (box). 1 cm^2 means 1 square and each side is 1 centimeter long. 1 cm^3 means 1 box and each side of the box is 1 centimeter long. For instance, if you make a square where each side is 5 cm, then you have a square that is 5cm^2. And 5^2 means 25, and that means you can fit 25 little 1cm^2 squares inside the big square that is 5cm^2. And if you have a cube (box) whre each side is 5 cm, then you have a cube that is 5 cm^3. And since 5^3 means 5x5x5 = 125, that means you can fit 125 little 1cm^3 cubes inside that bigger square that is 5cm^3.
cm^{2} is area. cm^{3} is volumn. A cube with 1cm long edges has sides with area 1 cm^{2} and a volumn of 1 cm^{3}.