In most logic textbooks and classes we hear it stated that in the formula $F \supset G$ (where $F$ and $G$ are syntactic variables), the connective `$\supset$' is a $\textit{"material conditional(or implication)"}$. Kleene in $\textit{Mathematical Logic (published by Dover)}$ p. 69f. offers some illumination. In order to distinguish it from (I take it) the $\textit{formal/logical implication(or conditional)}$, which is symbolized as $F \vdash G$ OR (given soundness and completeness) $ F \vDash G$ OR $\vDash F \supset G$ OR $\vdash F \supset G$, Kleene says something to the effect that formal/logical implication is expressed in the $\textit{metalanguage},$ while the material conditional (or implication) is expressed in the $\textit{object language}$.
So, the reason why we say that $F \supset G$ paraphrases a material conditional is that its truth value (as expressed in the object language) will "depend ordinarily on circumstances outside of logic, e.g. on $\textit{matters}$ of empirical fact." (Kleene, $\textit{Mathematical Logic}$ p. 70) See the connection between 'matters' and 'material'?
Kleene's point, as I understand him, is this: on the one hand, the formal/logical implication, since it is stated in the metalanguage does not concern itself with the interpretation of $F$ and $G$. It says in the $\textit{metalanguage}$ that $\textit{formally}$ or $\textit{logically}$ $F$ and $G$ are connected as $F \supset G$.
On the other hand, the material implication since it is in the object language, its truth value $\textit{depends}$ on the interpretation (or model) we give to $F \supset G$. In any model $F \supset G$ is true if either $F$ is false or $G$ is true; false otherwise i.e. when $F$ is true and $G$ is false. The only thing that the material conditional guarantees is that if $F$ is true, then $G$ is true. If $F$ is false, the material conditional tells us nothing about the truth value of $G$, which might turn out (on the basis of other $\textit{matters}$) to be either true or false.
In short, the differences are two: (1) material conditional is in the object language while the formal conditional is in the metalanguage; (2) the truth value of the material conditional depends on $\textit{matters}$ other than the formal relationship between $F$ and $G$.