The paper of Bryan and Leung ["The enumerative geometry of $K3$ surfaces and modular forms"](http://arxiv.org/abs/alg-geom/9711031/) 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 
$$
\sum\limits\_{n=0}^{\infty}N\_{g}(n)q^n=\frac{q}{\Delta(q)}\left(q\frac{d}{dq}G\_2(q)\right)^g
$$
where $\Delta(q)$ and $G\_2(q)$ are the modular forms:
$$
\Delta(q)=q\prod\limits\_{n=1}^{\infty} (1-q^n)^{24},\ G\_2(q)=-\frac{1}{24}+\sum\limits\_{n=1}^{\infty}\sum\limits\_{k|n}kq^n.
$$
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.



Curves on K3 and modular forms
The answer to your first question: "What is the relationship between $G\_2$ and $\Delta$?" is


$$q\frac{d}{dq} \log \Delta = -24G\_2 $$


where 


$$\Delta = q\prod\_{m=1}^\infty (1-q^m)^{24}$$ and 
$$G\_2 = -\frac{1}{24} +\sum\_{d=1}^\infty \sum\_{k|d}k q^d$$
(note that I've included the constant -1/24 which is in usual definition of $G\_2$ which you admitted). The above relation is easy to prove directly: take the log of the definition of $\Delta$, use the taylor series of log, rearrange the sum, differentiate, and out pops $-24G\_2$. 


To answer your second question of explaining this relationship in terms of counting curves, this can be done in the context of Gromov-Witten theory of the elliptic curve. Suppose that we wish to count the number of unramified covers of degree d of an elliptic curve $E$. We can either count connected covers, or possibly disconnected covers (and either way we should count each cover by the reciprocal of the number of automorphisms of the cover). Let $N\_{conn}(d)$ and $N\_{disc}(d)$ be the number of connected and possibly disconnected degree $d$ covers respectively. Let $$Z = \sum\_{d>0}N\_{disc}(d)q^d \quad\text{ and }\quad F=\sum\_{d>0}N\_{conn}(d)q^d $$ be the associated generating functions. Then the combinatorial relationship between possibly disconnected and connected covers yields 
$$Z=\exp{F}.$$


On the other hand, the number of connected covers is determined by covering space theory to be
$$N\_{conn}(d)=\frac{1}{d}\cdot\#\{\text{index $d$ subgroups of $\mathbb{Z}\oplus \mathbb{Z}$}\}$$


which one easily can prove is given by $\frac{1}{d}\sum\_{k|d}k$ by thinking about index $d$ sublattices or equivalently 2 by 2 integer matrices of determinant $d$. Thus we see that 
$$q\frac{d}{dq}F= G\_2 +1/24.$$ 


On the other hand, *possibly disconnected* covers are determined by their monodromy and this yields
$$N\_{disc}(d) = \frac{1}{d!}\cdot |\operatorname{Hom}(\mathbb{Z}\oplus \mathbb{Z},S\_d)|$$
where $S\_d$ is the symmetric group and the $1/d!$ prefactor is to remove the choice of a labelling of one fiber (necessary to get monodromy). The right hand side of the above equation can be computed with elementary group theory and one gets that $N\_{disc}(d)$ is given by the number of conjugacy classes in $S\_d$ which is in turn given by $p(d)$ the number of partitions of $d$. Thus $Z$ is the well known generating function for $p(d)$:
$$Z=\prod\_{m>0}(1-q^m)^{-1} = \left(\frac{q}{\Delta}\right)^{\frac{1}{24}}$$.


Putting this whole discussion together, we see that the relationship between possibly disconnected and connected covers of the elliptic curve, i.e. $Z=\exp (F)$ becomes


$$q\frac{d}{dq}\log \left(\left(\frac{q}{\Delta}\right)^{\frac{1}{24}}\right) = G\_2 +1/24$$


which is equivalent to the original relationship.


