Why descriptive power of context-free grammar is greater than regular expression?

Let the set of languages that can be represented by regular languages be $R$. And set of languages that can be generated by a context free grammar be $G$. Then surely $R \subset G$. For the proof that every regular language has a context free grammar that generates it, I encourage you to look up Michael Sipser's book on complexity. And for the second part, there are many languages which have a context free grammar generating them but are not regular. For example $L=\{ww^{reverse}\}$, where $w^{reverse}$ is reverse of the string $w$. How do we know $L$ has a context free grammar ( you can easily come up with a push down automata for it ), look up Sipsers. And look up pumping lemma in the book to prove $L$ is not regular.
To get an intuition, any regular language can be represented by a finite state automata $FSA$ and vice versa. And any context free grammar can be represented by a push down automata $PDA$ and vice versa ( again read the proof from Sipsers ). And in a $PDA$ in addition to the states ( like in $FSA$ ) you also have an infinite memory stack. Thus intuitively context free grammars have "power" greater than that of $FSA$.