One of the first to use letters as parameters in equations was Francois Viete (1540 - 1603). So use of letters in mathematics as variables, general numbers, and later for other things such as functions, operators, vectors, etc. started in renaissance.
One of the advantage of such use of letters is that we can write short, clear and unambiguous formulas e.g. Vieta's formulas for sum and product of solutions of quadratic equation. Before the use of letters in mathematics, mathematicians didn't write formulas, they would explain the formula with words. Pythagoras never wrote `c^2=a^2+b^2` but he said: "In any right-angled triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs."