Why can Solve solve this system of expressions but not a similar system?

Is this what you want?

Clear[a, b, c, d, e, f];
eqn = {p^2 == (a d - b c)^2/(Sqrt[(a + b) (c + d) (a + c) (b + d)])^2,
   s == (a + c)/(a + b + c + d), h == (a)/(a + c), f == b/(b + d)};

Solve[eqn, {p}, {a, b, c, d}]
(*
  {{p -> -(((f - h) Sqrt[-1 + s] Sqrt[s])/
      Sqrt[-f + f^2 + f s - 2 f^2 s - h s + 2 f h s + f^2 s^2 - 2 f h s^2 + h^2 s^2])},
   {p -> ((f - h) Sqrt[-1 + s] Sqrt[s])/
     Sqrt[-f + f^2 + f s - 2 f^2 s - h s + 2 f h s + f^2 s^2 - 2 f h s^2 + h^2 s^2]}}
*)

Even after fixing your error in specifying the equations (your equation for p involved the variables ad and bc instead of the products a d and b c), Solve[] still took its sweet time. We can help Solve[] out a bit by using GroebnerBasis[] as a preprocessor, which we can do since we have an algebraic system of equations (no transcendental functions involved). More precisely, we can remove the intermediate variables a, b, c, d like so:

neq = Simplify[First @
               GroebnerBasis[{p == (a d - b c)/(Sqrt[(a + b) (c + d) (a + c) (b + d)]),
                              s == (a + c)/(a + b + c + d), h == (a)/(a + c),
                              f == b/(b + d)}, {p, h, s, f}, {a, b, c, d}]]
   f^2 (p^2 (-1 + s) - s) (-1 + s) + h s (h - p^2 - h s + h p^2 s) -
   f (-1 + s) (-2 h s + p^2 (-1 + 2 h s))

Solve[] can now be used to solve for p:

Solve[neq == 0, p] // FullSimplify
   {{p -> ((-f + h) Sqrt[-1 + s] Sqrt[s])/Sqrt[(f (-1 + s) - h s) (1 + f (-1 + s) - h s)]},
    {p -> ((f - h) Sqrt[-1 + s] Sqrt[s])/Sqrt[(f (-1 + s) - h s) (1 + f (-1 + s) - h s)]}}

where two solutions were returned. Since (you can check this yourself!) h (1 - f) - (1 - h) f == h - f, the first solution is the desired one.

Unfortunately, the expression within the square root in the original source seems iffy, since (f (-1 + s) - h s) (1 + f (-1 + s) - h s)/(s (s - 1)) doesn't quite simplify to (h/(1 - s) + f/s) ((1 - h)/(1 - s) + (1 - f)/s). Checking which of these formulae is sensible to your application is something I'll leave for you to determine.