Historically, not much! Tang Soo Do is the traditional foundation while Taekwondo is the modern variation, but they are essentially the same art.
When the Koreans first started teaching empty handed fighting after WWII, they referred to their art under many different names. Tang Soo Do is the Korean pronunciation of "Karate." Then in the 1950s, against the will of the individual schools (called Kwans in Korean), the Korean government forced all of the Kwans to unify under a new name: Taekwondo.
Over time, this new art evolved into the national sport of Korea and was accepted into the Olympics as an official sport in 2000.
Many of the Kwan leaders felt that the art had abandoned it's roots as an effective martial art system in favor of competition training, and as such small pockets continued to call their art Tang Soo Do. The American Jidokwan Association was established to preserve the traditional foundation and practical self defense aspects of the art. Therefore, students earn rank in both Tang Soo Do and Taekwondo under the American Jidokwan Association!