American Tango and International Tango (together Ballroom Tango) came from Argentine Tango which is the original Tango. Ballroom Tango is what Americans think of when they think of "The Tango" or what they see in most commercials and movies such as "True Lies" and "Scent of a Woman" and old Rudolf Valentino movies. Argentine Tango can be scene in movies such as Assassination Tango and International movies such as "Tango" and "The Tango Lesson". Argentine and ballroom tango are totally different dances, each with its own distinct body of music and beauty.
Ballroom tango steps were standardized by dance studios in order to better facilitate judging at competitions. The steps have been relatively fixed in style for decades. Ballroom tango is very structured around memorized patterns of steps and is often choreographed and danced either as a performance or in competition. It tends to be very stylized and theatrical (think rose in the teeth).
However, Argentine tango is a constantly evolving dance and musical form, with continual changes occurring every day on the social dance floor in Argentina, Uruguay and in major tango centers elsewhere in the world even today. Argentine tango is primarily a social dance that include many styles and styles of music that allows one to dance it all night long without getting bored. Such dances as Vals, Milonga, Canyengue, Tango Nuevo; and styles as Milonguero style, Salon style are commonly danced and can be mutually exclusive. There are also styles created and named by prominent tango figures in the Tango world, such as Todaro style, Pulpo style, Gavito's Style.
Argentine Tango is still based heavily on improvisation. While there are patterns or sequences of steps that are used by instructors to teach the dance, even in a sequence every movement is led not only in direction but also speed and quality (a step can be smooth, pulsing, sharp, ... etc.). Though it can be choreographed and/or performed in a beautiful and dramatic way, its emphasis is on the potential for intimate connection and communication between partners.
Ballroom Tango:
Argentine Tango: