First, I found all pairs that satisfy the criteria:

Next, we can start making some deductions:
1 must connect both southwest and south, and 15 must connect both northeast and southeast. Since 15 only connects to {2,4,8}, and 8 is already used, 4 must go northeast of it and 2 must go southeast.
That gives this new grid:

Next,
1 must connect to 16 or 12, neither of which connects to 8. So that resolves more of the path. Additionally, 2 connects to 9 only out of the leftovers, which connects to 14 only, which connects to 3 only, which does not connect to the 13 it is adjacent to.
That resolves the rest of the path, but not the numbers:

For the numbers,
the 1 must have a 2-chain leading to 8. Out of the remaining numbers, that can only be 1-6-11-8. And finally, the 5 only connects to 12, which only connects to 7, which only connects to 10, and so the chain can be completed.
The final answer to the puzzle:

And to double-check the result:
The only extra lines we're concerned about are when an odd and an even number are adjacent, but not connected. These pairs are 11-14, 11-16, 7-16, and 1-8. None of these have both sum and difference prime, so the puzzle is solved.