I don't mean with translating at all. I mean in understanding language, including roots, structure, and derivations.
A language is not just a dictionary of words and their meanings. It is a highly complex model of communication with various rules described in grammar, orthography, pronunciation, sentence forming, etc. Now, the thing is that these rules tend to be more different for languages that are less related.
Two related languages may have an almost identical structure and a different vocabulary. Example: Italian and Spanish
Another pair of less related languages will have a different structure, grammar, etc. but at least they will have the same concepts - sentences, gender, numerals, and so on. Example: English and French
Less related languages would have a few quite different concepts. For example, most languages have singular and plural. Slovene has singular, plural and dual.
It's easy to imagine what kinds of differences may occur if we compare two completely unrelated languages. I don't have the knowledge to make the comparison, but I can assume that the list would be quite large for, say, Japanese and Portuguese.
Now, since you mentioned Latin specifically, I'll give one example why would it be useful. A native English speaker without knowledge of any other languages would (probably) have difficulties with noun case system present in some modern languages (e.g. most Slavic languages) since that concept is almost extinct in modern English. On the other hand, someone who had learned Latin wouldn't have too much trouble to figure it out. They would just say Oh, Родительный падеж, that's Genetivus, right And I don't mean just to know intellectually what that is - I mean to understand it, to grok it, if you will.
Understanding foreign concepts is the hardest thing when learning new languages. Now, why would Starfleet officers have to understand various concepts of languages and communication, I wonder? Could it have anything to do with all those first contact situations and the fine details of diplomacy? I would bet that they don't only learn Latin, but also other classical languages from Earth and other Federation planets, at least to some level of knowledge. Wouldn't it be strange if Starfleet trained its officers to attempt to understand alien communication without first training them to understand Earth languages?
Why would they bother? Both as far as Latin being a dead language, AND with the Federation having one of them Universal Translator doohikeys.
Universal Translators do have limitations. Just remember Darmok, and Jalad... at Tanagra.. Actually, the main plot of that episode is about a language with incredibly different concepts - a difference which their Universal Translator was unable to handle.