The paper of Bryan and Leung "The enumerative geometry of $K3$ surfaces and modular forms" provides the following formula. Let $S$ be a $K3$ surface and $C$ be a holomorphic curve in $S$ representing a primitive homology class. If $N_g(n)$ is the number of curves of geometric genus $g$ with $n$ nodes passing through $g$ generic points in $S$ in the linear system $|C|$ with any $g$ and $n$ satisfying $C^2=2(g+n)+2$ then $$ \sum\limits_{n=0}^{\infty}N_{g}(n)q^n=\frac{q}{\Delta(q)}\left(\frac{d}{dq}G_2(q)\right)^g $$ where $\Delta(q)$ and $G_2(q)$ are the modular forms: $$ \Delta(q)=\prod\limits_{n=1}^{\infty} (1-q^n)^{-24},\ G_2(q)=\sum\limits_{n=1}^{\infty}\sum\limits_{k|n}kq^n. $$ A relation between these two forms is well known in Number Theory. Unfortunately, I did not manage to find the relation and the first question is "What is the relation?". The main question is

How can one explain the relation between $\Delta(q)$ and $G_2(q)$ in the context of counting curves?

Some ideas are provided by the original paper of Bryan and Leung: these two modular forms correspond to numbers of (multiple) covers of a nodal rational curve and of an elliptic curve.