There were a number of copyright battles in which the US Soo Bahk Do Federation attempted to prevent the Tang Soo Do schools who had separated from the federation for one reason or another from using the name, forms, logos, intellectual property, etc. At the center was the name and logos, many of the schools who split off continued to use the name Moo Duk Kwan and the Federation logo. The Federation was interested in ensuring that there was a clear delineation between Soo Bahk Do and Tang Soo Do. Both of which being the art passed down from Hwang Kee, but following his death, the Federation was taken over by his son, HC Hwang, who maintains full control of the name, logos and intellectual property of his father. Many of the TSD schools who left have very high ranking masters who were close to Hwang Kee and went their separate ways to teach their interpretation.
In the end, it was ruled that the words "Tang Soo Do" could not be copyrighted, since they are as common and general as the term "Karate" (esp. since they are a direct Korean translation of the Hanja characters for Karate). Therefore, the name Soo Bahk Do was used in its place, but the Moo Duk Kwan remains sole property of the Federation (although Moo Duk Kwan Tae Kwon Do also uses the name, since they split off duruing the Kwan unification in 1940's/50's Korea).
As for the rest of the stuff, the logo can only be used by the federation in its exact configuration.....but the forms were ruled to be "dance movements" also general and common, such as walking. Since they are simply a combination of movements, they cannot be copyrighted.
SBD schools will, generally, teach a much more Chinese-influenced, softer art, while TSD schools will run a spectrum from Japanese-influenced hard to almost-SBD. SBD schools also teach additional form sets, the Yuk-Rho and Chil-sung sets created by GM Hwang, which most TSD schools will not.
In my experience, this is a minor misconception. Though Soo Bahk Do does have some chinese influence, I have
personally seen this to be more of an addition to the art rather than a completely altered style. The Tang Soo Do schools that I was in did teach a very hard, Japanese based kind of style....but the Soo Bahk Do schools that I have trained with have taught basically the same curriculum, simply with the addition of some softer techniques. This "softer side" is mainly found within the Chil Sung and Yuk Ro forms, although the Federation has added new Basic movements to reflect these softer, chinese style movements. The bottom line is that the harder side remains.
(e ship yuk - not intending to disagree, just expand a bit)