Note that's it's not civil matters but matters related to government action (from say, basic rights to labor disputes for State employees or citizenship issues).
One difference among many: the Cour de Cassation does not issue decisions, exactly; it can only hold or break a decision from a lower court. If it chooses to break the lower court's decision (casser=to break) then the case is sent back to said court to be decided again, with new guidance from the upper court.
That's also typically how the U.S. Supreme Court works, except for those few oddball cases where it has original jurisdiction. The losing party from the lower court files a petition for certiorari (judicial review). The Supreme Court may grant it, hear the two parties' arguments, and reach an opinion which is sent back to the lower courts, who are then responsible for resolving any remaining questions.
