Gunboat Diplomacy is a substantial improvement over Charismatic Leader: it provides twice as many Influence Points toward earning Envoys and also gives you Open Borders with all city-states. It can help you win a Domination Victory by giving your army and navy much greater freedom of movement, or a Diplomatic Victory (in Gathering Storm) by helping you earn the Envoys you need to become the Suzerain of multiple city-states and receive more Diplomatic Favor.
Gunboat diplomacy – or, in Teddy’s words, the “Big Stick” policy – is the pursuit of foreign policy objectives through conspicuous demonstrations of overwhelming military force, with the threat of open hostilities underlying the display by the more powerful nation. Roosevelt himself described it as “the exercise of intelligent forethought and of decisive action sufficiently far in advance of any likely crisis.” The term came into popular usage during the period of European imperialism, as the colonial powers used the presence of warships to force concessions (e.g. Perry's Black Ship, the Paknam Incident, etc.).