Well, let's start out with what's not the difference - they're both corn.
Polenta is typically made from coarsely groung yellow corn meal. It's boiled very slowly until the cook has what many call "corn meal 'mush'." After cooked polenta has cooled and hardened, it can be sliced, sauteed, or grilled.
In many parts of Mexico, and especially in northern Italy, polenta is a beloved dish that's served daily, often topped with meat, fish, pasta sauce, cheese (this is true especially in Mexico), or vegetables. Many Italians love their polenta torta, a layered dish that's limited only by the imgination of the cook and is reminiscent of lasagna.
Polenta can be combined with many different ingredients, so the final product can be either sweet or savory.
Grits, served most often in the American South, are "coarsely ground pieces of dried corn moistened into a mealy paste." According to NPR, their role in Southern culinary culture is almost mythic.
Historians believe that grits provided food for the first English settlers in Jamestown, Virginia, and later helped many Southerners survive the deprivations of the Great Depression.
There is a difference between corn grits, which include both the hull and the germ of the grain, and "true" hominy grits.
In order to make hominy, one starts (no, not with a can), but with field corn. The dried corn kernels are soaked in a solution of baking soda, lime, or wood ash (also known as "lye water") for one to two days. When the kernel's shell falls off, the kernel absorbs the water and swells to more than twice its size (ouch!). The kernels are then rinsed several times, dried, then finally ground into grits. The grind can be coarse, medium, or fine.
It is, in fact, this alkaline soaking process, which also adds to the nutritional value of the food, that differentiates grits from polenta.
The same soaking process is used to make masa harina, which is the key ingredient in corn tortillas. Due to their altered chemistry because of the soaking, both grits and tortillas helps to prevent pellagra, a disease caused by niacin deficiency.
Corn is beloved in so many cultures, but often, when someone says "polenta" or "grits," people not from those cultures turn their noses up or think of something that's perhaps way too "down home" for them. This really isn't true. Both polenta and grits can be contributors to very sophisticated dishes when used by a creative and imaginative cook.