The easiest answer is the style: the visual look and feel of Japanese animations vs. American animations, mostly evident in the design of human characters. The distinctive large eyes with numerous reflective highlights and detailed color are the main hallmark of anime, along with small noses and mouths generally denoted by minimal lines. (Even certain styles that favor unrealistically broad, generous mouths depict them using minimal lines.) The style itself uses many angles and flowing, attenuated lines. Things such as eyelashes, hair, and clothing are depicted in more fine detail. The color often uses more variants and shading, with greater attention paid to non-outlined highlights and shadows to add more depth.
In contrast, American animation either falls into attempts at comic-book style "realism" (as realistic as that can get, anyway) or grossly exaggerated, comical cartoon characters with rounded, highly exaggerated features. There's usually less detail, focused instead on using tricks of style to imply the detail in more subtle, understated fashion, and less attention to shading instead of solid block colors save for in dramatic scenes that require it.
Where American animation may seem to lack in that aspect, though, it makes up for it in the amount of animation done. American animation includes a great deal of original animated motion - some of it used cyclically, but still animated painstakingly frame by frame. In contrast, anime uses a lot of cheats: long scenes in which only the mouth of a character (and maybe a few strands of hair) moves during delivery of key information, or depicting rapid motion with a character frozen in an action pose against a swift-moving, stylized background that requires little animation. Often use of dramatic still shots against patterned backgrounds with a few moving emotive symbols will accompany a monologue. Both styles reuse shots and sequences, but Japanese animation tends to be a little more obvious about it. This is why Japanese anime is sometimes labeled as "lazy" by American animators.
The style element goes a bit further than just drawing styles, though. American animation tends to use straight-on camera shots, less concerned with cinematic angles and dramatics than with clearly depicting the events, though there are exceptions to that rule. Japanese animation will often make use of exaggerated angles, perspectives, and zooms to intensify the mood of a scene and show actions to extreme effect.
The largest difference, though, is in content and audience. In America, for the most part animated cartoons and films are considered to be for children, and are targeted for that audience. In Japan, anime can be for children or adults, and some Japanese imports have caused some interesting surprises when parents discovered their children had something of a more mature nature. Also, the idea of what's appropriate for children and appropriate for adults can differ between the two cultures, and what's appropriate for a ten-year-old in Japan may not be considered appropriate for a ten-year-old in America. Most of that can be explained by cultural differences, and an American watching Japanese anime may notice cultural references or context clues from the locations that wouldn't be present in American animations.
Beyond that, though, the differences aren't really so great. Both seek to tell a story in an animated medium, using both digital and traditional methods. Both use exaggeration to emphasize the emotion in character actions, as well as other tricks such as anticipation, well-timed music, and squash and stretch. Both follow the principles of animation and require an absolute dedication to the craft. In the end there's really no one which is better; it's just a matter of taste and preference.