For the _summation_ symbol $\Sigma$, the use of the capital Greek letter for "s" is due to the use of Latin in Early Modern Europe as universal language; see summa : (mathematics) sum, summary, total.
The first usage is due to Leonhard Euler in 1755; see Institutiones calculi differentialis, page 23 :
> Quemadmodum ad differentiam denotandam vsi sumus signo Δ, ita _summam_ indicabimus signo $Σ$.
Similarly for the long $s$ :
> Leibniz favored the name _calculus summatorius_ and the long letter $\int$ as the symbol. Bernoulli favored the name _calculus integralis_ and the capital letter $I$ as the sign of integration. [...] Leibniz and Johann Bernoulli finally reached a happy compromise, adopting Bernoulli's name "integral calculus," and Leibniz' symbol of integration.