What does this structure (有 … 才有 …) mean?
They both mean the first part is the necessary (under many situations, also sufficient) condition of the second part. 

有A才有B means only if A exists, then B exists. Here 有 has real meaning.

只有A才B means only if A then B. Here 有 does not have its real meaning, but is part of the conjunction.

Example:

有国才有家 (Only if the nation exists, does family exist)

只有有国，才会有家 (same meaning, grammatically correct, but rarely seen)

只有国，才有家 (wrong)