For you to understand the difference a bit better, let's say that you would say
This is a cheaper book - Αυτό είναι (ένα) φθηνότερο βιβλίο
This one is cheaper than that one - Αυτό είναι φθηνότερο από εκείνο
(Preferably not "This is" the cheaper book. It's not commonly used like this in English, since you're only comparing it to another one)
Τhis is the cheapest book - Αυτό είναι το πιο φθηνό βιβλίο. (from all the other books).
I hope I helped ^.^
We don't have a special construction for that in Greek, so the translation would theoretically be the same as the given sentence, but "το πιο φθηνό" generally means "the cheapest". I guess you need a certain context to translate it as "the cheaper", as in the examples you mentioned.
A bit like in English, comparatives can be formed analytically (with a separate word: more expensive, πιο ωραίο) or synthetically (with an ending on the word: cheaper, καλύτερο).
The superlative is generally the definite article plus the comparative; thus, there are cases where the superlative has a πιο in it and cases where it doesn't, just as in English there are both "the most expensive book" with "most" and "the cheapest book" without it.
- πιο φθηνό = cheaper
- το πιο φθηνό = the cheapest
Modern Greek forms the superlative by using the definite article plus the comparative.
As I understand it, it can also mean "the cheaper (book)", if there are exactly two.
But if there are more than two, one would say "the cheapest book" in English, but in Greek it would still be το πιο φθηνό βιβλίο or το φθηνότερο βιβλίο.
"This book is cheaper" = Αυτό το βιβλίο είναι πιο φθηνό.
"This book is the cheapest one" = Αυτό το βιβλίο είναι το πιο φθηνό.