You ask a good question, because back when I dabbled in tkd in college, we had a club in college and the club changed instructors the former being tkd the latter being tang soo do, and he said we would all be able to keep our ranks and all despite being an alledgedly different style.
Myself, I got turned off and left the class. i would be curious how tkders feel about this.
I studied Tang Soo Do for quite a few years. I hold the rank of 1st Dan in the art. The words "Tang Soo Do" mean The way of the hand as influenced by the Tang Dynasty of China (Tang referring to the Chinese Dynasty, Soo meaning Hand, and Do meaning Way). Some of the differences between Tang Soo Do and Tae Kwon Do: Tang Soo Do can trace its lineage back over 2,000 years. It uses the legs, feet, arms, hands, elbows, knees, and any other part of the body as a weapon. It is a "defensive" art, meaning we do not attack unprovoked. We defend ourselves and our loved ones. It teaches the student to live a healthy, harmonious life threw rigorous training, both mentally and physically. Tae Kwon Do is a combination of several martial art styles in Korea that all joined together under 1 umbrella and called it Tae Kwon Do. Tae Kwon Do is mainly a "sport" but does have several practical self defense applications, if you find the right teacher. Today, there are several "styles" of Tang Soo Do and Tae Kwon Do. The schools can be as different as night and day, so if you do not like one school, do not let it turn you off to the art.
Tang Soo Do more closely resembles Japanese karate in that it uses forms taken pretty much wholesale from Japanese and Chinese systems. Also, if you watch them, Tang Soo Do students seem more stiff in their movements, more akin to karate students. Taekwondo students tend to be more fluid and more likely to use continuous action.
Taekwondo over the last few decades has made a conscious effort to get back to classical Korean methods of practice, while Tang Soo Do has tried to stay closer to the methods of Hwang Kee. I'm not criticising it by the way.
My understanding is that originally there was no difference. In 1966 there was a split. Tae Kwon Do Moo Duk Kwan is very similar to the the original Tang soo do Moo Duk Kwan. Since then different federations have evolved in both Styles and different hyungs, competition rules and emphasis imposed resulting in 2 different styles.
It is quite obvious that there will be a slight bias based on which "camp" you are a member of. So I will do my best to remain objective here, but I am a member of the Soo Bahk Do/Tang Soo Do camp (although, those two camps are slowly growing further apart).
There is first, of course, the difference in the naming. That is quite obvious and has already been addressed.
Next, you have the difference in history. There are of course claims that various Korean martial arts trace their histories back thousands of years and are based on a variety of "ancient styles" including Tae Kyon, Subak, Hwarang, and others. For reality's sake though, based on the available verifiable, unbiased, and objective history, the styles of martial arts practiced in Korea today trace their roots back to the end of the Japanese occupation. I realize that this topic can set off a thread of its own and many people are very passionate about the age and history of their style, but I think that we can all agree that during the Japanese occupation, a great deal of Korean history and hard documentation was lost, so following the occupation, the basically started over using what was remembered and hidden. The major historical difference was that when the original 9 Kwans united, under the umbrella, the Moo Duk Kwan (Tang Soo Do, Hwa Soo Do, Soo Bahk Do at different times) did not unite with them. There was at least one other kwan which did not unite, although I have heard conflicting stories as to whether it was Jido Kwan or Song Moo Kwan, both of which have since joined the Tae Kwon Do umbrella. The Moo Duk Kwan, to the best of my knowledge is the only of those original 9 Kwans who are still surviving today under their original set up and naming. Of course, there are division under Tae Kwon Do who maintain their original Kwan name as well. The Moo Duk Kwan is also still run and maintained by the Hwang Family (Hwang Kee passed it along to his son, H.C. Hwang).
Last, of course is the difference in style. There is a big problem here though. Across Tang Soo Do and Tae Kwon Do alike, there is a HUGE difference in styles. In fact, there are styles of TSD which are almost inditinguishable from certain styles of TKD and vice versa (aside from Hyung).
There is of course, the "sport" aspect of TKD which TSD really doesn't have much of. There are tournaments in TSD, but none of the level of the olympic style sparring found in some TKD schools. Not all TKD is sport based though, there are certainly styles of TKD which are much more traditional. In my observation, there does seem to be more Japanese and Chinese influence in TSD, the chinese side particularly in Soo Bahk Do. TKD seems to use much faster, lighter movements, intended to shorten the distance to the target. TSD seems to me to have much more use of the hips than most TKD that I have seen.
Another very minor difference is in naming - not of the art, but of the movements. I'm talking about Poomse vs Hyung; kong kyuk vs chirugi; etc.
To answer the question though......It depends. Tell me which style specifically you're looking at, under which organization and we can get a lot more specific!
Off topic a bit, it's mentioned in the Soo Bahk Do Dae Kam that the Jidokwan joined Moo Duk Kwan in the early 60's. They left the Moo Duk Kwan sometime later, and Yoon Kwai Byeong kept his Jidokwan faction separate from Taekwondo, renaming it Tae Kuk Kwan, and passed it on to Jung, Dong-Hoon.
Well I'am a 1th degree black belt under Ji Do Kwan TKD, mi sambunim always told us TKD evolved from So Bak/Tang Soo Do, I quit TKD at 22 and returned to it 1.6 years back under Hwarang Tae Kwon Do (in Veracruz Mexico) my actual sambunim was a Moo Duk Kwan TKD for 25 yars then he split form MDK and founed his own dojan (Hwarang).
Something we (taekwondoings) have been discussing in the TKD forum is th lak of self defense technikes and the proliferation of olimpic TKD, some of us did not like this.
I quit TKD 18 years back cause it was focused in the olimpics and discarded SD and traditional TKD.
I tought TSD was more martial oriented and fosused more in SD than TKD.