There has been gestural communication among deaf people as long as there have been deaf people. The earliest reference to it is from Plato, almost 2500 years ago; it has also been documented throughout western history. Sign language first became recorded in the 1600s, though, with documentation beginning of the actual methods it employs, rather than just references to it.

Different countries use different sign languages for the same reason they use different spoken languages. No one "invented" sign languages - they are natural languages. So just as spoken languages differ from country to country, so do signed languages.