Suppose $m$ is a positive integer. A quantity of interest is

$$
H_m = \liminf_{n\to\infty} \left(p_{n+m} - p_n \right)
$$

The twin prime conjecture, is, of course $H_1 = 2$, the the prime k-tuples conjecture of Hardy and Littlewood asserts that $H_2 = 6$, $H_3 = 8$ and so on. Goldston-Pintz-Yildirim showed that under the Elliot-Halberstam conjecture, $H_1 < \infty$, a result that Yitang Zhang  has famously recently established unconditionally.

It is my understanding that the methods of GPY were essentially restricted when dealing with $m>1$. One of their non-trivial results was that $g_n = p_{n+1} - p_n$ satisfies $g_n = o(\log p_n)$, or in other words that $\liminf \frac{g_n}{\log p_n} = 0$, which obviously trivially follows from Zhang's result.

However, it is my understanding that even under the full Elliot-Halberstam conjecture, GPY were unable to prove even their weaker result for $m\geq 2$. Maynard and Tao (see http://arxiv.org/abs/1311.4600) have shown that in fact, that $H_m < \infty$. Apparently, this work only required the classical Bombieri-Vinogradov theorem, not even the variation of that Zhang's proof uses.

So my question is this: what shortcoming did the GPY approach have that did not allow it to extend to $m>2$, and how has Maynard's approach solved this problem?

PS: I apologize if this question is somehow inappropriate for Mathoverflow. I am an undergraduate trying to read up on these recent results by myself as much as I can, and I figured this would probably be the best place to ask my question.