Well, how different they are first depends on what organization of Tang Soo Do you came from.
Second, let's clarify what the difference between TSD and MDK is. When TSD was founded in 1947, it was called Tang Soo Do Moo Duk Kwan. The Moo Duk Kwan was one of the original 9 kwans, Tang Soo Do was the style of MDK. Over the last 50 years, the Moo Duk Kwan has very slowly fragmented. Over the course of time, the official "parent organization" changed its name from Tang Soo Do Moo Duk Kwan to Soo Bahk Do Moo Duk Kwan - largely for copyright reasons. You see, the words Tang Soo Do are a literal translations of the chinese characters for Karate. Soo Bahk Do is a less generic term. When that change was made, the senior masters who framgented from the parent organization for a variety of reasons, took the name Tang Soo Do for their organizations.
So fast forward to today. Right now, the United Stated Soo Bahk Do Moo Duk Kwan Federation is still alive and well in the US. The World Moo Duk Kwan is also alive and well and has many child organizations in many countries. The US SBD Fed and the Wrold MDK is still run by the Hwang Family, namely Hwang Kee's (the founder of TSD) son, HC Hwang.
As the other oganizations have fragmented off of the WMDK, each has taken on slight stylistic differences. It is the same art, same basic style, but with minor variations.
The MAJOR variations that you will find in the Moo Duk Kwan:
1) Standardized ho sin sul and ill soo sik
2) The use of the yuk ro and chil sung hyungs
3) A slightly softer, chinese approach than many of the other organizations who have stayed more Japanese in application
You will find most of the same hyung, kibon dongcha, etc, but it will take a while to get caught up with the new hyung, hosinsul and illsoosik if you choose to move to MDK.
One caveat....there is nothing wrong with the other organizations. You will hear that they "no longer have access to the founder" through his son....but honestly, they are all headed by very senior MDK practitioners who are excellent martial artists. The quality of the school is determined by the instructor, not the organization.
I would be happy to answer any more questions on the subject. I have trained with several different organizations still within the TSD style, so I have some different perspectives.