Why Fiorentina is commonly translated as "佛罗伦萨" instead of "费奥伦庭那"？
Chinese 佛罗伦萨 fololunsa
Italian Firenze  [ˈfire̞nt̪s̠e̞]
English Florence /ˈflɒrəns/ 
Latin flōrentīna /fjo.renˈti.na/ etymology from flōrēns (“flowering”).


As above, 佛罗伦萨 is translate from English pronunciation.

Why not from Italian (翡冷翠,still in use today for some literary effect) or from Latin(费奥伦庭那 as you typed)?


  中国人谙英文者多,而习小语种者少,欧美许多非英语国家地名往往从英语转译,不是直接从该国语言译>来,例如慕尼黑由英文的转写(transliteration)munich而来,而非来自原名München[1]


详情请看翡冷翠 vs. 佛罗伦萨：译音及其原则, It's very long.

Why 亚特兰大 not 阿塔兰塔 ?

In fact, I think 鱫ai脦te兰lān踏tà is most close to English pronounciation. Completely transliterating results some weird words like 鱫脦兰踏. And for the perspective of specification, something like 英汉译音表 is a good choice.[2] . And the previous link is also helps a lot in this question.

Reference


http://www.cnki.com.cn/Article/CJFDTotal-YWJZ201002040.htm
(sorry you probably cannot download it)
picture comes from 世界人名翻译大辞典>附录二:译音表


notice, I am not professional.