Both of these terms, ακριβότερο and ακριβέστερο, are derived from the same root. Ακρίβεια means both 'expensiveness' (lol) and 'accuracy'. From that you get ακριβός/ακριβή/ακριβό (expensive) and ακριβής/ακριβής/ακριβές (accurate).
In general, derivative words are formed following certain rules, lots of which you get instinctively as a native speaker and lots of others that you learn during school grammar lessons. In this case, I would simply say that the comparative adjectives follow closely their respective root words and this is (fairly) obvious in their sound (-ό- vs -έ-).
I did see that. My question remains, as I am trying to understand the nuances of the language, and the correct use of articles is one of the most difficult aspects of every language I've studied. Does πριν in an ago/last sense REQUIRE the use of an indefinite article? Can πριν with a definite article ONLY be translated as before x?
I think το προηγούμενο , as suggested by Stergi3, succinctly translates to "last" as you describe. And I think the definite article is required; in Greek, just saying "προηγούμενο καλοκαίρι" is like just saying "preceding summer", which isn't correct in English. Youd say "THE preceding summer".
In English, the "the" in "last summer" is implied (i.e. "THE last summer" means the exact same as "last summer"). The omission works in English, but I can't give a satisfying reason why.
And I've gone off track...but on the same note, and to answer your question, "καλοκαίρι πριν" is like saying "summer ago", which also doesn't make sense in English. You have to quantify it. One summer ago? Two? Forty? From my point of view Ένα isn't really an indefinite article here, it's the specifying number.