The War of Jenkins Ear did not result in significant gains for either the British or the Spanish, so in terms of “who won” it was inconclusive. It did result in a weakening of Spain's overall grasp of the Americas vis a vis Britain's.
In the aftermath of the War of the Austrian Succession which was mainly fought and settled in Europe there were still ongoing clashes between Britain and Spain, in particular, raids on each other’s trading ships in the Americas. Captain Robert Jenkins claimed that in 1731 he had lost his ship, the Rebecca, and his ear, to an attack by Spanish coast guards. By 1738 these claims were used within Britain as a political cause celebre to attack and pressure Prime Minister Robert Walpole and in October 1739 he declared war on Spain.
Admiral Edward Vernon attacked and captured the Spanish fort of Porto Bello, Panama, in 1739 but in 1740 was defeated at Cartagena, Colombia. Also in 1740, the Governor of Georgia, James Oglethorpe was defeated attempting to capture St Augustine, Florida.
On 9th June 1742, the Spanish attacked St Simon’s Island, Georgia and were repulsed by Governor Oglethorpe’s troops at the battle of Bloody Marsh.
In 1743, James Oglethorpe again assaulted St Augustine and was again defeated, resulting in his withdrawal from Florida.
Here is a more detailed view of Vernon's campaign