To find the difference equation from the transfer function, we first recognize that H(z) is the ratio of the output to the input i.e. Y(z)/X(z). Divide the transfer function through by z-2 and multiply both sides by the denominator to obtain:
Y(z)*(1 + 0.67565z^-2) = X(z)*0.16217*(1 + z^-2)
Taking the inverse Z transform yields a difference equation:
y[n] = 0.16217x[n] + 0.16217x[n-2] - 0.67565y[n-2]
WinPOI: Pitfalls of the Impulse yields a difference equation of:
y[n] = +1.0000*x[n] +0.0000*x[n-1] +-1.0000*x[n-2] +0.0000*y[n] +-0.6760*y[n]
The POI equation does not contain the gain, but aside from that the two equations match perfectly.