The difference is in the basic counting function they considered. In the GPY paper "Primes in tuples I", the counting function used is the following

$$\displaystyle \sum_{n=N+1}^{2N} \left(\sum_{i=1}^k \theta(n + h_i) - \log 3N \right) \Lambda_R (n; \mathcal{H}_k, l)^2$$

In particular, the key is to show that this sum is positive for all large $N$, which means that some parts of the sum must have positive contribution. The weight $\Lambda_R (n; \mathcal{H}_k, l)^2$ is obviously positive so this implies that $\displaystyle \left(\sum_{i=1}^k \theta(n + h_i) - \log 3N \right)$ must be positive, which implies that for infinitely many $n$ the tuple $n, n+h_1, \cdots, n+h_k$ must contain at least two primes.

Note that here the counting function involves $\theta(n) = \log n$ for $n$ a prime, and zero otherwise. This is based on the premise that this function is easier to work with analytically than just the characteristic function on the primes.

However, Maynard was able to circumvent this technical difficulty and was able to work with this counting function instead

$$\displaystyle \sum_{n=N+1}^{2N} \left(\sum_{i=1}^k \chi_{\mathbb{P}}(n+h_i) - \rho\right)w_n.$$

Here $\chi_{\mathbb{P}}$ is the characteristic function on the primes, and $\rho$ is a positive weight. Because Maynard was able to get a positivity result for this counting function, it immediately implies that when it can be proved that the sum is positive, there exists infinitely many $n$ such that at least $\rho + 1$ many of $n, n+h_1, \cdots, n+h_k$ are prime.

The difference is in the basic counting function they considered. In the GPY paper "Primes in tuples I", the counting function used is the following

$$\displaystyle \sum_{n=N+1}^{2N} \left(\sum_{i=1}^k \theta(n + h_i) - \log 3N \right) \Lambda_R (n; \mathcal{H}_k, l)^2$$

In particular, the key is to show that this sum is positive for all large $N$, which means that some parts of the sum must have positive contribution. The weight $\Lambda_R (n; \mathcal{H}_k, l)^2$ is obviously positive so this implies that $\displaystyle \left(\sum_{i=1}^k \theta(n + h_i) - \log 3N \right)$ must be positive, which implies that for infinitely many $n$ the tuple $n, n+h_1, \cdots, n+h_k$ must contain at least two primes.

Note that here the counting function involves $\theta(n) = \log n$ for $n$ a prime, and zero otherwise. This is based on the premise that this function is easier to work with analytically than just the characteristic function on the primes.

However, Maynard was able to circumvent this technical difficulty and was able to work with this counting function instead

$$\displaystyle \sum_{n=N+1}^{2N} \left(\sum_{i=1}^k \chi_{\mathbb{P}}(n+h_i) - \rho\right)w_n.$$

Here $\chi_{\mathbb{P}}$ is the characteristic function on the primes, and $\rho$ is a positive weight.

Edit: As pointed out below, the main difference between this and the above counting function isn't the use of the characteristic function $\chi_{\mathbb{P}}(n)$ over $\theta(n)$ (because looking at the range of $n$ be considered, one can see immediately that $\log N < \theta(n+h_i) < \log(2N)$ ). The difference, then, was that Maynard was able to obtain a positivity result for (comparing to the original counting function) $$\displaystyle \sum_{n=N+1}^{2N} \left(\sum_{i=1}^k \theta(n+h_i) - \rho \log3N\right) w_n,$$ where $\rho$ is now allowed to be any positive number. The details of how he is able to obtain these results (or at least a sketch of the main ideas) is given in the answer below, which admittedly is much deeper than this one.