Ancient Greek has two words that are translated as life in English: zōē and bios. What is the difference between them? What are their cognates in other Indo-European languages?
These two words are actually cognate with each other; they show the differing reflexes in Greek of labiovelar consonants. In the noun ζωή zōē and the corresponding adjective ζωός zōos, the initial ζ developed from a cluster with the semivowel y; the noun βίος bios shows the normal development of PIE *gʷ before the full vowel /i/.
The Online Etymology Dictionary's entry for the (English) prefix bio- also lists an assortment of cognates in other branches of Indo-European:
Sanskrit jivah "alive, living;" Old Persian *jivaka- "alive," Middle Persian zhiwak "alive;" Old English cwic, cwicu "living, alive;" Latin vivus "living, alive," vita "life;" Old Church Slavonic zivo "to live;" Lithuanian gyvas "living, alive," gyvata "(eternal) life;" Old Irish bethu "life," bith "age;" Welsh byd "world")
It also has the following note on the meaning:
The correct usage is that in biography, but in modern science it has been extended to mean "organic life."
This is supported by the definition of βίος in Henry George Liddell and Robert Scott's A Greek-English Lexicon, edited by Sir Henry Stuart Jones:
life, i. e. not animal life (ζωή), but mode of life
(accessed from the Perseus Digital Library on 15 February 2016)
From Proto-Hellenic *ǰṓw-ō, from *gʷyṓw-ō, from Proto-Indo-European *gʷíh3w-oh2.
In Attic Greek inherited /w/ was lost.
Piotr Gąsiorowski explains in the comments here how βίος and ζωός developed from the same Proto-Indo-European root:
*gʷih₃w-ó- ‘living’ and *gʷíh₃w-e/o- ‘to live’. The change *-ih₃- > *-jō is an inner Greek development (parallelled by similar changes in Tocharian and Armenian). It’s known as Francis–Normier’s law, or laryngeal breaking. Not everybody accepts it, but they really should. The ‘life’ root alone would be evidence enough for it. [...]
The Greek development of the inherited labiovelars is complex and differs from dialect to dialect, but the normal (expected) Attic reflex of *gʷ before *i and *ī is b, not d (though *gʷj yields z). Therefore, both βίος and ζωός are regular.
Regarding their meaning/usage during the "Classical" and "Hellenistic" period in non-technical literature, I would say ζωή refers the opposite of "death" in terms of existence, and is used for animals, plants, etc., with common overtones of transcendence, whereas βίος refers to human lifestyles and activities, which by metonymy would also relate to things need to sustain such "life."
In Christian theology the usage of the original greek words in the bible give way to the churches to interpret the message for congeragation members. FE. Jesus said he is the way, the truth and the life (ζάω). In this case ζωή differs from bios. ζωή refers to 1 to live, breathe, be among the living (not lifeless, not dead). 2 to enjoy real life. 2a to have true life and worthy of the name. 2b active, blessed, endless in the kingdom of God. Bios refers to just life as opposed to not being alive.