To put it simply, 多谢 is a form of 唔该, but 唔该 is not necessarily 多谢。

多谢. duoxie in Mandarin or “door-jay” in Cantonese, is literally “many thanks”. It’s a phrase purely used for expressing thanks or gratitude. It’s normally used in these situations:

1. Upon receiving a gift
2. Upon receiving a favor (that you never requested for)

唔该 (“Mm-goi” in Cantonese) is a pure Cantonese phrase and not used in Mandarin. It’s literal meaning is “should not” or “ought not”. It’s actually an abbreviation of the phrase “麻烦你真唔应该” meaning “should not be troubling you” or “so sorry to trouble/bother you”. It’s a phrase used under these circumstances:

1. To express thanks upon receiving a service from another person, example : when someone open a door for you (ie you are thanking the person for performing a service for you or carried out a task that you could have done it yourself).
2. To express “please” or “gratitude” when asking the other person to do something for you; example : asking someone to pass you something; 唔该攞支笔俾我, (please take a pen for me).
3. As a substitute for “hello” with a “please” meaning, like “I beg your pardon” or “excuse me” in English, when speaking to a stranger for the purpose of enquiring about something, example : asking for direction; 唔该，请问你中环系边度 (Hi or please or excuse me, can you tell me where is Central District)?