Followup question, my last name is August and I can trace myself back to Lithuania as "Augustinavicious" (anglicized). Would there be any connections between Augustus and Augustinavicious?
My personal guess is "august" being a loanword, but I'm amazed it made it's way to Lithuania.
In most cases, English does not do this. Only in a minority of cases did English assimilate (almost always from French) the practice of spelling or pronouncing a Latin name after a fashion which reflected a Vulgar Latin form (such that Fr. Auguste where Eng. Augustus and Fr. Tibère where Eng. Tiberius). In an even fewer cases, a fully Anglicised form is adopted for its own sake. But this is usually outside the strictly historical/scholarly sphere.
Broadly speaking, this question is one of those which, as they often do, seems to demand consistency and unbreakable rules of historical linguistics in a fashion which generalises across 1500 years of linguistic history. And such answers are seldom on offer. Ultimately, English Anglicised some few Latin names and failed to Anglicise most others 1) because English is phonotactically and orthographically diverse, and very regularly allows exceptions to its phonotactic and orthographic rules, which have changed quite a bit over the course of its history 2) English was not the scholarly language of Britain until modernity, creating plenty of opportunity for popular and scholarly variants of Latin names, with the situation further complicated by the contributions of Anglo-Norman French (which while not the language of the common man, was the predominant language of power centres of society, throughout the Middle English period).
One generalisation which can be made is that biblical and ecclesiastical terminology was the Latin vocabulary (and to some extent, but usually indirectly, Greek) most immediately nativised, in English. In Old English, we do get Petrus and Paulus rather than Peter and Paul. But as the names most relevant to the common man (who does not need to know the history of the Roman Republic, but needs to know how to conduct his prayers), Anglicised forms of the names of the disciples and church fathers are among the first to appear. Some Latin place names relevant to contemporaneity (rather than merely classical history) also receive this treatment before the Old English period has ended, for I would argue largely grammatical reasons. Writing "Roma" without the use of appropriate Old English case suffixes (especially dative and genitive) threatens to make its use rather awkward and grammatically obscure. So we see "Romeburg" instead. Place names of little popular relevance do appear in their native forms (e.g., "Antiochia" in the Anglo-Saxon Chronicle), even in spellings which disregard Anglo-Saxon orthographic conventions (in this case, "ch" is not a digraph native to Old English).
There is no one factor, nor easy answer. English has had a complex history. And some tendencies, in historical linguistics, simply impose themselves inconstantly.
The simple answer is that those names that end in 'an' are (usually) adjectives.
In Roman naming customs, when a man was adopted by a different family, they would not simply discard their original family name. Instead, they would often change its form into an adjective in order to indicate that, although they were not a member of their old family any more, they were still 'of' that family.
A good example is Augustus himself. He was born Gaius Octavius (Octavius being his family name). When he was adopted by his uncle, Julius Caesar, he changed his name to Gaius Julius Caesar. But in order to distinguish himself from his uncle, he was often known as Octavianus, which essentially means 'one from the Octavian family'. This is why he is known as Octavian today, when referring to the period before he adopted the title Augustus.
This is ultimately the case for most other well-known Romans with this naming convention. The emperor Trajan (Marcus Ulpius Trajanus) had the name 'Trajanus' passed down from his father, likely because of an adoption earlier in his family history.
The reason we translate these names to end in 'an' is because that's just how that kind of adjective (indicating that someone is from something) usually ends in English. For example, a Roman is someone from Rome. We get this convention from Latin: Roman comes from 'Romanus', which also means someone from Rome. The change wouldn't occur for a name like Titus because Titus is simply his name, rather than what is essentially an adjective that can be translated.
Sources:
http://www.sjsu.edu/faculty/watkins/caesaraugustus.htm
http://klio.uoregon.edu/im/re/trajan1.htm
http://www.therthdimension.org/AncientRome/RomanNaming/romannaming.htm