As others have said, in practice both are used for the material conditional, and I certainly wouldn't want to say that someone is 'wrong' in using the one symbol rather than the other, but personally I have my reasons for separating between the two:
I use $\rightarrow$ for the material conditional, so that I can use the $\Rightarrow$ for logical implication. For example, I would use $P \land Q \Rightarrow P$ to make the meta-logical statement that the logic statement $P$ is logically implied by the logic statement $P \land Q$. Likewise, I use $\leftrightarrow$ for the material biconditional, and $\Leftrightarrow$ to express logical equivalence. For example: $P \leftrightarrow Q \Leftrightarrow Q \leftrightarrow P$ expresses the meta-logical statement that the logic statement $P \leftrightarrow Q$ is logically equivalent to the logic statement $Q \leftrightarrow P$ (of course, some use $\equiv$ to express logical equivalence, but I have also seen $\equiv$ used to express the material conditional ...)
In short: people use different symbols, and that's just fine as long as you make clear what they mean and how you use them, but to me the single horizontal line signals something about the syntax of logic, while the double line to me signals something semantical. Indeed, in my eyes this distinction mirrors the distinction between $\vdash$ and $\vDash$ where $\vdash$ is about purely syntactical derivations, and the $\vDash$ is about semantical implication.