There is a very big difference between moving a two-legged animal and moving this four-legged animal. The ballroom four-legged animal we are going to talk about is truly a freak of nature. It doesn't exist naturally in nature and its movement has continued to evolve throughout the history of Ballroom dancing.
The two-legged animal has a single head to control the movement of one body. The four-legged animal has one head (the leader) and one head (the follower) to control the simultaneous movement of two bodies. Communication between the two halves is what makes partnership dancing work. The method of movement that the leader uses to move is the primary communication link. Ballroom dancers (four-legged animals) use one method of movement when they are dancing Latin/Rhythm dances and use another method of movement when they are dancing Standard/Smooth dances. This is what gives each dance style its unique characteristics and feelings.
The following descriptions of movement, presumes a proper connection of the partners through the dance frame and the correct relationship of the dancers to each other.
The Latin/Rhythm style of movement is quite similar in almost all of the different Latin/Rhythm dances. Of course there are always exceptions (Samba being one of them). The Latin/Rhythm "leg" extends from the bottom of the rib cage to the floor. This style of dancing employs a "leg" action called Latin/Cuban Motion. The steps are taken implementing "ball-flat" footwork. This allows Latin/Rhythm dances to be danced in a confined area. In Latin/Rhythm style dancing, the leg/foot always precedes the body's weight transfer. Rotation of the supporting leg/hip initiates the movement of the free leg/foot in preparation of the weight transfer. The movement of the body weight is achieved by lifting through the ankles. This action allows the use of both legs during the weight transfer. The body weight transfer passes from the supporting foot, through split weight, then on to the new supporting foot. The "body" (from the bottom of the rib cage and up) remains still.
Most of the different Standard/Smooth style dances have unique movement characteristics. In this style of dancing, the "leg" extends from the hip socket to the floor and the body (from the hip socket up) remains still. Footwork implementing "heel" steps is what allows the Standard/Smooth dancers to progress around the floor. Fundamentally, this style of dancing implements a power and glide type of technique. For example, in Waltz the power is generally supplied from the supporting leg during the "one beat" and the glide happens during the two and three beats. The person moving forward is in the best position to supply the power for the movement. This could be either the leader or the follower, but the leader is still in control of when the power is applied. The "rise and fall" in Slow Waltz, the "staccato" movement in Tango, the "very even movement" in Slow Foxtrot, the "extreme sway" in Viennese Waltz, etc., all give each Standard/Smooth dance a unique feel and characteristic.
As you can see from the descriptions above, the Latin/Rhythm and Standard/Smooth four-legged animals move very differently. The Latin/Rhythm dances are similar in their movement because they all employ Latin/Cuban Motion. The International Latin dances use International style Latin/Cuban Motion. The American Rhythm dances use American style Latin/Cuban Motion. Very generally, International style Latin/Cuban Motion is executed with a straight leg and American style Latin/Cuban Motion is executed with a bent leg. Also, very generally, in International style Standard dancing the feet close and in American style Smooth dancing the feet pass. The follower in both of these styles of dances isn't just dead weight being pushed around by the leader, she participates in lead, and she adds to the lead, she powers the lead; she enhances the lead, etc. She holds up her half of the two-headed four-legged animal.
This is a very high level brief description of the differences between these types of ballroom dancing. Hopefully this will give you a hint of how they differ and peak your interest to explore movement more closely.