How do PrP mutations lead to prion disease?
If the amino acid sequence of both proteins is the same,
  what determines whether the synthesized protein will take the
  disease-causing tertiary structure or the normal one?


PrP-C and PrP-Sc do indeed have the same primary structure. However, they differ in secondary and tertiary structure. The protein can take more than one shape (conformation), where the likely conformation spaces taken by the protein are due to chemical and physical interactions. Many proteins can change their shape, often this is not well understood in detail. However, we know that the conformational space of some proteins allows for several, stable conformations of a single protein; the factors which stabilize it in a conformation or cause it to switch are difficult to understand in vivo.

PrP-Sc tends to accumulate in compact, protease-resistant aggregates. Circular dichroism shows that normal PrPC had 43% alpha helical and 3% beta sheet content, whereas PrPSc was only 30% alpha helix and 43% beta sheet. Although the exact 3D structure of PrPSc is not known, it has a higher proportion of β-sheet structure in place of the normal α-helix structure. The end of each fiber acts as a template onto which free protein molecules may attach, allowing the fiber to grow.

Prion proteins therefore propagate accumulation of more and more prion proteins.


  Is it due to post-translational modifications?


No.


  How do prions, which are void of nucleic acids,
  integrate into the genome of a newly infected host by reverse
  translation, and become a familial disease?


They do not enter the genome through reverse translation. The DNA sequence itself is mutated and the mutation drifts across generations for as long as it is being passed on through the germline.