Words borrowed directly from Ancient Greek usually use т, б and е/э for θ, β and η: теология, термометр, библиотека.
Works borrowed through Bysantine Greek (translated from Bysantine sources) use ф, в and и: Фёдор, Фермопилы, Афины, Фивы.
Some words changed their spelling from Bysantine to Ancient Greek with time (вивлиофика -> библиотека, апофика -> аптека etc.). The spelling of such words was not stable at the time, this is why Mitya spells it the Bysantine way while the modern spelling is the Ancient Greek way.
In the modern language, new words are borrowed using Ancient Greek reading and the reading of existing loanwords is stabilized, so I'm not aware of words in the modern language which are read both ways (one of the readings would be considered obsolete).
Some common names of the same origin, spelt the same in other languages, are spelt differently in Russian: for instance, Bethlehem, Israel is spelt Вифлеем (like in Bible translations), while Bethelehem, PA is spelt Бетлехем.