I found that the following expressions have the same meaning:
Das ist mir Wurst.
Das ist mir egal.
But I’m still not sure if they can be used in same situation and if they express the same exact meaning. I want to understand the nuance between the two expressions, and it would be also great to know the etymology of the first one.
"Das ist mir wurscht/wurst" means the same as "Das ist mir egal". Neither are polite, although wurscht/wurst is a bit more colloquial.
An example:
A: Wollen wir heute abend zum Italiener oder zum Griechen?
B: Das ist mir egal. / Das ist mir wurscht/wurst. Hauptsache es gibt was zu Essen!
Duden and Redensarten-Index both list wurscht and wurst. Wiktionary says wurscht is used outside of northern Germany. I'm from southwest Germany and I only know wurscht, but I never thought of it as dialect, because I would never pronounce the sausage "Wurst" with "sch"!
Whereas Duden (1, 2) says "etymology unknown", Redensarten-index presents two possible etymologies:
Die Redensart "das ist mir wurscht", die wohl besagt: Es ist ganz egal, ebenso wie egal ist, was in die Wurst hineinkommt. Eine andere Deutung bezieht sich auf die zwei gleichen Enden der Wurst (Alles hat ein Ende, nur die Wurst hat zwei). Es ist also gleichgültig, an welchem Ende man sie anschneidet. Die Redensart stammt aus der Studentensprache Anfang des 19. Jahrhunderts
The saying "das ist mir wurscht", which probably says: It does not matter, just as it does not matter with what a sausage is filled. Another interpretation refers to the two equal ends of the sausage (all has an end, only the sausage has two). It is therefore indifferent at which end they are cut first. The style of speech comes from the student language at the beginning of the 19th century.
Important note: Despite its origin from sausage (Wurst). The wurst in das ist mir wurst/wursch is written in small letters.
EDIT and copy of the answer of Torsten Link to the difference with his friendly permission (see comments)
Das ist mir egal is the official (though not very polite) way to say: I don't care.
Das ist mir wurst is a colloquial way to say the same. You would usually not use this phrase in any official conversation (e.g. with your boss).
An often found answer to the etymology of this expression is, that a sausage ("Wurst") has two endings, so it doesn't matter from which side you start to cut or eat.
There are some more explanations as the expression can be found BC. But this is the one you can find the most.
Das ist mir egal is the official (though not very polite) way to say: I don't care.
Das ist mir wurst is a colloquial way to say the same. You would usually not use this phrase in any official conversation (e.g. with your boss).
"Wurst" here does not refer to sausage, but probably comes from Old English "wiersa" and similarly old Dutch "wers" which means worse (also related is the word war) or not good. It probably originally had only a bad connotation as in "it does not interest me at all or I don't like it" The German word "wurst" can also be used personally: Dieser Mensch ist mir wurst! It is not capitulated and not a noun but an Adjective in this case! https://www.duden.de/rechtschreibung/wurst We have a similar sazing in Afrikaans: Vir botter (butter) of vir wors (sausage) = for better or worse, which has nothing to do with sausage, but is borrowed from English.
