\begin{align} . \end{align} \begin{align} \text{Why is } \emptyset \neq\{\emptyset\}? \end{align}
\begin{align} . \end{align}
\begin{align} . \end{align} \begin{align} \text{Why is } \emptyset \neq\{\emptyset\}? \end{align}
\begin{align} . \end{align}
Two sets are equal only if they have the same elements.
Part of your confusion is about what elements a set is considered to have. In mathematics containment means direct containment while in every-day speak you allow for an indirect meaning. In everyday language you look into containers inside too.
Take for example your kitchen drawer contains a box of chocolates. In everyday language you'd probably say that there's chocolates in the drawer, but in math language one says there's only a box in the drawer (that box in turn happens to contain chocolates).
When you've then eaten all chocolates, but leaves the box you may consider the drawer to be empty since there's no longer any chocolates. But in math language it's still a box in the drawer (and that box happens to be empty).
Taking this analogy you consider if a drawer with an empty box has the same contents as a drawer with not even an empty box. In math language this is not the case since the first drawer contains a box while the other doesn't (that the box is empty doesn't change this fact).
The first set ($\emptyset$) has zero elements, the latter ($\{\emptyset\}$) has one element. Therefore they cannot be the same.