Mercer's theorem and the Moore-Aronszajn theorem are often reproduced in varying forms, but in essence, the difference and similarity between them are roughly as follows:
Let $k: X\times X\to \mathbb{R}$ be a kernel with $k$ and $X$ satisfying some required properties, such as $k$ being positive definite and continuous, and $X$ being compact. Then you can define an integral operator $T_k$ on the $L_2$-functions on $X$:
$$
T_k(f)(x) = \int k(x, y) f(y) d\nu
$$
for some appropriate measure $\nu$.
Mercer's theorem
Mercer's theorem creates a feature map $\Phi_M$ from $X$ to a Hilbert space by first computing the eigenfunctions $\phi_k$ of $T_k$ and then mainly just evaluating all of them at the point $x$, thus giving a map $\Phi_M: X\to \ell_2$, using the standard inner product on $\ell_2$.
Moore-Aronszajn theorem
The work of Moore and Aronszajn on the other hand takes a different approach. They consider for each $x\in X$ the function $f_x: y \to k(x, y)$ and consider the Hilbert space of continuous functions $C(X)$ with the inner product given by $(f_x, f_y) = k(x, y)$. Thus, the feature map $\Phi_{MA}$ is here given as $\Phi_{MA}: X\to C(X), \Phi_{MA}(x) = f_x$.
Comparison:
Even though the approaches look different, it can be shown that they are equivalent, in the sense that both map the kernel to the scalar product, i.e. both $\Phi_M$ and $\Phi_{MA}$ satisfy:
$$
k(x,y) = \langle \Phi(x), \Phi(y) \rangle,
$$
and that both Hilbert spaces are isometric.
The above information has been extracted mainly from:
Cucker, Felipe, and Steve Smale. "On the mathematical foundations of learning." Bulletin of the American mathematical society 39.1 (2002): 1-49.