The plastic tubs in many inexpensive dishwasher models are relatively porous, so stains from food and water tend to settle into the tub's surface and stay there, and because the plastic is usually light-colored, the stains stand out. Stainless steel is not as porous and resists staining; it is also odor-resistant for the same reason. Some mid-range dishwasher models, however, are equipped with gray or dark-colored plastic tubs; these tubs are also resistant to visible staining, and they are typically less expensive than models with stainless steel tubs.
Steel is, of course, extremely durable, and stainless steel tubs will last a long time. Plastic is also very durable, however. Plastic won't rust, and even the most inexpensive plastic tub will likely outlast the rest of the dishwasher. You'll probably replace the entire unit before you have to worry about the tub wearing out, even if it's made from inexpensive plastic.
Stainless steel tubs are sometimes thought to be better at muting the sound produced by the dishwasher, and it's true that more expensive models that feature stainless tubs tend to be quieter than inexpensive models. The overall design of the dishwasher is the most important factor in its sound-reducing capabilities, though, and a well-insulated tub will be quieter than a poorly insulated one, regardless of what material it's made from.
Stainless steel has an advantage over plastic when it comes to the drying cycle. Steel reflects more heat back into the tub, so heated drying cycles work more effectively. Steam in the tub also tends to condense on the tub walls instead of on the dishes, and the walls dry relatively quickly compared to the dishes; this reduces the drying time, and it's possible that you might not even need to use the heated drying cycle.