Bash shell scripting ( [[ ]] conditions)

A question was asked recently what is the difference between "[ ]" and "[[ ]]" when used in a conditional statement in bash shell scripting.

As I found making use of the [ ] provides what is known as a "simple" test.

if [ "one" = "one" ]; then
echo "one is one"
fi

Where as the [[ ]] provides for a compound test in addition to having conditional operators that are more typical in other languages.

if [[ "one" == "one" && "two" == "two" ]]; then
echo "Normality restored"
fi

Operators like && || and the like are not possible in the simple test