As discussed here, names for the same city in different languages tend to be just variant pronunciations. By contrast, a country that is known by (even roughly) the same name in most languages is the exception rather than the rule. I have some theories about this, but no real facts.
Theory 1: Country names often incorporate common nouns, city names more rarely. For example, it's easy enough to translate "Great Britain" into "Grande-Bretagne", or "United States" into "Estados Unidos", but what can you do with "London" or "Chicago". This theory would explain why New York is often called "Nueva York" in Spanish, but leaves Germany rather mysterious. Also, I've noticed that Spanish-language names for American cities are never translated. Los Angeles isn't called "Les Anges" or "die Engel". San Francisco has a Chinese name but it's 旧金山, Jiùjīnshān, meaning "Old Gold Hill", not 亞西西的方濟各 (Yàxixī de fāngjìgè — Francis of Assisi) which would be the literal translation.
Theory 2: City names, and cities, are stable, whereas countries, their borders, and their names are more fluid. Damascus and Rome have existed for most of recorded history; Syria and Italy are fairly modern inventions. When a country flickers in an out of existence, its name may survive in other languages and be (inconsistently) applied to a successor country. This theory would explain Germany, but it certainly doesn't explain Syria and Italy, which seem to be called pretty much the same thing by everybody.
Theory 3: Country names are changed for purely political reasons, and those reasons may not be respected by other countries. This would explain Burma and the Koreas, but nowhere else that comes to mind. It also raises the question of why, when city names are changed for political reasons (St Petersburg, Gdansk), those changes are tracked, fairly assiduously, even by countries that disagree with the underlying politics.
These theories are all post hoc and none of them explain China or Egypt. The hypothesis might not even be true but just a symptom of my patchy knowledge of the subject. Any suggestions or links to scholarship on the subject are welcomed.
Best Answer
I think this question starts from a false premise. Both country names and city names are sometimes translated, sometimes not. I am no expert in this matter, and I won't offer a tentative explanation as when or why this happens. But from my standpoint it's easy to observe that plenty of city names (as well as country names) change depending on what language you are speaking. A couple of examples off the top of my head:
So I wouldn't attempt to explain why country names and city names behave differently. I don't think they do.