Like McHenry stated, it really depends on the school/association. I practice Moo Duk Kwan TSD, but my Kwan Jang Nim left Korea before all the additions of the Chinese flavor of today's Moo Duk Kwan Soo Bahk Do. We practice the old Shotokan hyungs with Korean flavor. Pyung Ahns, Naihanchi, Bassai, Chinto, Kang Song Koon are the form sets. We wear the cross wrap dobahks, not the V-necks.
A good martial art focused (rather than sport focused) TKD school looks very similar a martial art focused TSD school other than the dobahks and form sets, IMO.
The closer the association with H.C. Hwang a TSD school is, the more Chinese influence they are likely to have. Much of the Chinese form sets were not integrated until much later (I've heard as late as the 1980's).
Kong Soo Do ("the way of the open hand") and Tang Soo Do ("the way of the Chinese Tang hand") are both translations of the word Karate Do. The majority of all of the original schools in Korea were heavily based in either Shotokan or Shudokan Karate. The change of the name to TKD and the creation of new form sets and dobahks were attempts to distance themselves from the Japanese, who once occupied their country.
The basic techniques themselves, at that time, were still identical. The forms are a different subject, however. The newer form sets were created with no thought of boon hae (bunkai), so if applications are of interest, the older Shotokan forms are of more importance, IMHO. As time went on, things did change, due to the focus of the art. So if you find an instructor lineage that focused on the old TKD or the old TSD, you'll find they are nearly identical in techniques.