Over a period of eight years resentments built, Walpole attempted to avoid war, the English subjects wanted war. In 1739 Jenkins testified before Parliament, displaying the reputed severed ear. War against Spain was declared October 23, 1739. This went on for some time, and lots of people died, including Colonial Americans who for the first time were engaged as British soldiers in a foreign war.2
This war had significance in the New World, for at the time the English controlled Georgia, whereas Florida was owned by Spain. In raids on each other battles were fought up and down the south coast of Georgia, defended by Oglethorpe, not the least of which was Bloody Marsh after the Spanish landed on St. Simons Island.3
Jenkins War was soon folded into the War of Austrian Succession, all of which ended with the Treaty of Aix-la-Chapelle in 1748. The St Johns River was unofficially recognized as the border between Florida and Georgia, and other issues, including trade by the British in Spanish America, were settled by the Treaty of Madrid in 1750.2
1. In addition to piracy and severed ears, it should be noted that underlying these hostilities were trade issues, including the "right" of the English to trade slaves in Spanish America, control of territory, and the general bellicose proclivities of mankind which seems to have persisted throughout history, and which still persists to this day.
2. After eight years between the severing and the exhibiting of the ear, it could well have been a pig's ear, for all I know.
3. There really is a lot of interesting reading on the subject if one wishes to follow some of the links provided below.