From the standpoint of biological classification, alligators and crocodiles are members of two distinct families. They are both members of the order crocodilia. Thus, all alligators and all crocodiles are crocodilians.
Note an additional biological family in Table 1, the gavialidae. The gavial, or gharial, is a fish eating (piscivorous) saltwater crocodile native to northern India. The gharial is a highly endangered species, with a known number of extant individuals only a little over 200. As can be seen in the photo below, the gharial has a very long, narrow snout specialized for catching and consuming fish. Very different morphology than other crocodilians indeed.
Other differences between alligators and crocodiles are the placement of teeth and jaw plan. Alligators have a larger upper jaw than their lower jaw. Thus, with jaws closed, an alligator's lower teeth are not visible (part A below). Crocodile's upper and lower jaws are approximately the same width (part B), and when closed, the upper and lower teeth interlock.
Alligators and crocodiles both have integumentary (skin) sensory organs know as dermal pressure receptors. These highly sensitive receptors can detect the faintest of water perturbations, assisting them in locating and tracking prey. These receptors are distributed differently in alligators and crocs. Both have DPRs located in the skin of their upper and lower jaws. However, crocs also have DPRs distributed about most of their body's surface area.
There are behavioral differences as well between the two families. Crocodiles, in general, are more aggressive than alligators. Alligators will often retreat from humans, where crocodiles may have a greater propensity to attack. However, one has to be careful in overgeneralizing, as the speciation of crocodilians is very complex and vast. Thus, depending upon the particular species, there will be many differences in temperament, aggressiveness, as well as size.