Let $P$ be a $\sigma$-finite measure and $L^2(\Omega, \mathcal{F}, P)$ be the corresponding function space where $\Omega$ is the domain and $\mathcal{F}$ is the $\sigma$-algebra. Let $\mathcal{C}\subseteq\mathcal{F}$ be a sub $\sigma$-algebra, then what is the differences and connections between the two spaces: $L^2(\Omega, \mathcal{F}, P)$ and $L^2(\Omega, \mathcal{C}, P)$?

The reason I am asking this question is that during the proof to show the existence of conditional expectation using the Riesz representation theorem, it is claimed that $L^2(\Omega, \mathcal{C}, P)$ to be a closed subspace of $L^2(\Omega, \mathcal{F}, P)$. I understand that a function $f$ that is $\mathcal{F}$-measurable is not necessarily $\mathcal{C}$-measurable, but the other direction does not necessarily hold, but is there any other more intuitive properties that distinguish functions in $L^2(\Omega, \mathcal{C}, P)$ from functions in $L^2(\Omega, \mathcal{F}, P)$?