What is the difference between $x$ and $\{x\}$ when $x$ itself is a set?

Think of the brackets as a bag you put things in. Then $\{1\}$ is a bag containing the number $1$. But $\{\{1\}\}$ is a bag containing a bag containing the number $1$. So two bags, one inside the other. These are different. Physically different if you think real paper bags.

$$\{1\} $$ is a set whose the unique element is the integer $1$

$$\{\{1\}\} $$ is a set whose the unique element is the set $\{1\} $.

Well if you have $x=\varnothing$, then $0=\#x\neq \#\{x\}=1$. So clearly both sets are not the same.

Edit: With $\#S$ I refer to the cardinality of a set $S$, i.e. in the finite case the number of elements in $S$.