What are the differences between the Hilbert space and real space representations in chemistry?
A Hilbert space is a kind of linear vector space.

In chemistry which encounter it when quantum mechanics, when we can represent wavefunctions by their contributions from different orthonormal single particle states.  It is these single particle states which we build up out our atomic orbitals (AOs).

Example:

Consider a two level systems - minimal basis $\ce{H2}$:

Each atom has a $1s$ AO on its hydrogen: $\phi_1(\textbf{r})$, $\phi_2(\textbf{r})$.  These AOs aren't orthogonal though:
$$
\iiint d^3\textbf{r}\ \phi_1^*(\textbf{r}) \phi_2(\textbf{r}) \ne 0
$$but we can construct orthonormal states:
$$
\psi_\pm(\textbf{r}) = \frac{1}{\sqrt{2}}\left(\phi_1(\textbf{r})\pm\phi_2(\textbf{r})\right)
$$
such that like functions integrate to $1$ and functions of unlike functions vanish:
$$
\iiint d^3\textbf{r}\ \psi_\pm^*(\textbf{r}) \psi_\mp(\textbf{r}) = 0 \\
\iiint d^3\textbf{r}\ \psi_\pm^*(\textbf{r}) \psi_\pm(\textbf{r}) = 1
$$.
These two orthogonal function can be represented as state vectors:
$$
\psi_+(\textbf{r}) = \pmatrix{1\\0},\; \psi_-(\textbf{r}) = \pmatrix{0\\1}
$$
This defines a Hilbert or vector space, where our single particle states $\psi_\pm$ are our vectors and the inner product of two vectors, $\psi_i$ and $\psi_j$, is defined by the integral over all space:
$$
\langle\psi_i\vert\psi_j\rangle = \iiint d^3\textbf{r}\ \psi_i^*(\textbf{r}) \psi_j(\textbf{r})
$$
and is equivalent to the dot product of two state vectors.

In a electronic structure (specifically DFT) code, a "real space code" means using atom orbitals as basis functions rather than plane waves.

The Hilbert space described above is complete - it describes all possible wavefunctions which can be made from the AOs included.  If you included the infinite number of hydrogenic wavefunctions on each atom you would be able to have an exact description of all possible wavefunctions for $\ce{H_2}$ - as the hydrogenic wavefunctions form a complete set, instead you include as many as practical.

An alternative complete set of functions is the infinite set of all plane waves of all frequencies.  In cases where you are performing calculations on a periodic system, such as a solid, plane waves can be a more natural choice as they themselves are also periodic.  In a box of width $L$, the set of waves with wavelengths:
$$\lambda = \frac{2L}{n}$$
for all integral n, form a complete set.  They are more commonly described by their wavevectors, $\textbf{k}$, a vector in what is called reciprocal space, which describes the momentum of the wave.

In DFT, when calculating integrals over density, in a plane wave code you manipulate the density in terms of wavevectors, $\textbf{k}$.  In a real space code, you compose your density from Kohn-Sham orbitals in real space and calculate integrals of the density over quadrature grids in real (Euclidean) space.

Hence "real space" is used as a descriptor, for DFT codes that work with AOs and basis functions defined on atoms in Euclidean space, in contrast to "plane wave" codes which used basis functions defined in position space.