I have been reading about Runge-Kutta methods, particularly the "classical" fourth-order method. When it is talked about, the 3/8th rule is often mentioned. For example, in this document, the classical method is said to be more popular, but somehow the 3/8th rule is more precise.
The wikipedia page of the list of Runge-Kutta methods also states that the classical rule is more notorious, even though both were presented in the same paper.
Why exactly is one method more popular than the other?
Because the formulas are simpler, easier to remember, and the method quicker to implement on-the-fly.
Wilhelm Kutta (1901) gave the 3/8 method \begin{array}{c|cccc} 0&\\ \frac13&\frac13 \\ \frac23&-\frac13&1\\ 1&1&-1&1\\ \hline &\frac18&\frac38&\frac38&\frac18 \end{array} first as a symmetric solution $b_1=b_4$, $b_2=b_3$ to the 4th order equations/coefficient parametrizations.
Next he considered methods that generalize the the Simpson integration rule where he gives the examples $$\begin{array}{c|cccc} 0&\\ \frac14&\frac14 \\ \frac12&0&\frac12\\ 1&1&-2&2 \\\hline &\frac16&0&\frac46&\frac16 \end{array}~~ \begin{array}{c|cccc} 0&\\ \frac12&\frac12\\ 0&-\frac12&\frac12 \\ 1&-\frac32&\frac32&1 \\ \hline &0&\frac46&\frac16&\frac16 \end{array}~~ \begin{array}{c|cccc} 0&\\ 1&1\\ \frac12&\frac38&\frac18\\ 1&\frac14&-\frac14&1 \\\hline &\frac16&-\frac16&\frac46&\frac26 \end{array} $$ and then as the last 4th order example what we know as the classical Runge-Kutta method (which really should have been named Heun-Kutta method), where he highlights that this is the only 4th order method that is compatible with both his and the approach of Karl Heun (1900).
