The main difference between the two is that wantons usually do not have any type of dough placed inside the wonton wrapper. This is usually filled with vegetables and meat. Dumplings can also e filled with different ingredients that are mixed like dough, and they will usually be in the form of balls. You can actually distinguish if you are looking at a wonton or a dumpling based on their appearance.
Dumplings and wontons are usually served with different sauces depending on how they are cooked. Some would love to dip them in some sweet Asian sauce while there are also others who would dip their wontons or dumplings in soy sauce and chili.