If $B$ is **necessary** for $A$ to be true, then $$\neg B\Rightarrow \neg A$$
that is, if $B$ is not true, then $A$ cannot be true. This is the same as saying that, if $A$ is true, then $B$ must also hold. Note this says nothing about the statement $B\Rightarrow A$.


On the other hand, if $B$ is **sufficient** for $A$ to be true, then

$$B\Rightarrow A$$

that is, if $B$ holds, then $A$ also holds. Similarily, this says nothing about the statement $A\Rightarrow B$.

This is why, in general, if $B$ is a **sufficient and necessary** condition for $A$, then $A\iff B$.