Why is it that some nouns use their case endings with adjectives while others, like this, do not?
It is used here. However, if you mean the -tun (or the nuncation on Ta Marbuta at the end of مدينة), this is actually a malfunctioning in the text-to-speech machine that Duolingo is using here. The phrase above should be: Madínatun 3arabiyyatun, OR Madínatun 3arabiyyah (the vowel on the last letter in the last word in the sentence can be dropped since the sentence is over and nothing is expected after it).
Well, in English translation of the name it is written as Medina, the Arabic name is Al-Madínah, AND, it is an abbreviation for Al-Madínah Al-Munawwarah المدينة المنورة (the enlightened city) - so called after being a base for the spread of Islam (its old name was Yaþrib يثرب). So, the name reached English via the abbreviation and not via the full name of the city or town.
Bonus: In Malta also, there is a town called Mdina which is derived from the same Arabic word مدينة.