What is the second-order approximation error of the Heun method, which averages the slope at the beginning and the end of the interval?

$$y(x+h) = y(x) + \frac{h}{2} \left( f(x, y(x)) + f[x+h, y(x) + hf(x, y(x))]\right) \tag{8.33}$$

Let's look at the Taylor expansion of the last term:

$$ \begin{align} f[x+h, y(x) + h f(x, y(x))] &= f(x, y(x)) + h \frac{\rm d}{\rm dh} f\left[x+h, y(x)+h f(x, y(x))\right]_{h=0}\\ &= f(x, y(x)) + h\left[\frac{\partial f}{\partial x} + f \frac{\partial f}{\partial y}\right] + \mathcal{O}(h^2) \end{align} \tag{1} $$

Combining equations (1) and (8.33) provides:

$$ \begin{align} y(x+h) &= y(x) + \frac{h}{2} \left( f(x, y(x)) + f(x, y(x)) + h\left[\frac{\partial f}{\partial x} + f \frac{\partial f}{\partial y}\right] + \mathcal{O}(h^2)\right)\\ &= y(x) + h f(x, y(x)) + \frac{h^2}{2} \left[\frac{\partial f}{\partial x} + f \frac{\partial f}{\partial y}\right] + \mathcal{O}(h^3) \end{align} \tag{2} $$

which is the same second order term as the midpoint method. Only the third order error will show differences between the two methods.