The primary difference between the two styles of waltzes is the timing and tempo. The original waltz is slower than the Viennese waltz. Both waltzes use three-quarter time, but the Viennese waltz can be up to four times as fast in regards to the tempo. This faster tempo is known as canter timing, and has two steps per measure rather than the slow waltz's three steps per measure.
The choreography is different between waltzes and Viennese waltzes. The original waltz consists of turns and linear movements. The Viennese waltz is sometimes referred to as the rotary waltz, because the choreography consists mainly of turns and rotations. You can spot the difference between the two styles of waltzes based on how many turns you see the partners performing around the ballroom floor. Inside and outside twirls are common choreographed steps that are notable to Viennese waltzes.
In a regular waltz the dance partners have moments when they can pause in certain positions, such as the man dipping the woman back, or the two staring at each other. The Viennese waltz is too fast paced to permit such pauses. Besides turning around and around, the dance partners may have a brief moment where they open their connection to release their arms to the side --- while still moving --- and then reconnect to their original closeness again.
The tone of the slow waltz is more romantic than the upbeat Viennese waltz. A Viennese waltz is too fast to allow for emotional connection between the dancers to play a part of the performance. Since the slow waltz lets couples take their time along the dance floor, the romantic or dreamy ambiance becomes woven into the steps. Viennese waltzes, on the other hand, cast a different type of ambiance --- one of pleasure and purpose.