What's The Difference Between Memorial Day and Veterans Day ?

The Veterans Administration FAQ page answers the question this way:

A. Many people confuse Memorial Day and Veterans Day. Memorial Day is a day for remembering and honoring military personnel who died in the service of their country, particularly those who died in battle or as a result of wounds sustained in battle. While those who died are also remembered, Veterans Day is the day set aside to thank and honor ALL those who served honorably in the military - in wartime or peacetime. In fact, Veterans Day is largely intended to thank LIVING veterans for their service, to acknowledge that their contributions to our national security are appreciated, and to underscore the fact that all those who served - not only those who died - have sacrificed and done their duty.

Let's start with Memorial Day, since it's what is being celebrated today.  From another VA page:

Three years after the Civil War ended, on May 5, 1868, the head of an organization of Union veterans — the Grand Army of the Republic (GAR) — established Decoration Day as a time for the nation to decorate the graves of the war dead with flowers. Maj. Gen. John A. Logan declared that Decoration Day should be observed on May 30. It is believed that date was chosen because flowers would be in bloom all over the country.
The first large observance was held that year at Arlington National Cemetery, across the Potomac River from Washington, D.C. . .

. . .  By the end of the 19th century, Memorial Day ceremonies were being held on May 30 throughout the nation. State legislatures passed proclamations designating the day, and the Army and Navy adopted regulations for proper observance at their facilities.
It was not until after World War I, however, that the day was expanded to honor those who have died in all American wars. In 1971, Memorial Day was declared a national holiday by an act of Congress, though it is still often called Decoration Day. It was then also placed on the last Monday in May, as were some other federal holidays.

And now Veteran's Day.  Again, borrowing from the VA website:

World War I – known at the time as “The Great War” - officially ended when the Treaty of Versailles was signed on June 28, 1919, in the Palace of Versailles outside the town of Versailles, France. However, fighting ceased seven months earlier when an armistice, or temporary cessation of hostilities, between the Allied nations and Germany went into effect on the eleventh hour of the eleventh day of the eleventh month. For that reason, November 11, 1918, is generally regarded as the end of “the war to end all wars. . .”

. . . In November 1919, President Wilson proclaimed November 11 as the first commemoration of Armistice Day . . . 

. . . The original concept for the celebration was for a day observed with parades and public meetings and a brief suspension of business beginning at 11:00 a.m.

The United States Congress officially recognized the end of World War I when it passed a concurrent resolution on June 4, 1926 . . .

. . .An Act (52 Stat. 351; 5 U. S. Code, Sec. 87a) approved May 13, 1938, made the 11th of November in each year a legal holiday—a day to be dedicated to the cause of world peace and to be thereafter celebrated and known as "Armistice Day." Armistice Day was primarily a day set aside to honor veterans of World War I, but in 1954, after World War II had required the greatest mobilization of soldiers, sailors, Marines and airmen in the Nation’s history; after American forces had fought aggression in Korea, the 83rd Congress, at the urging of the veterans service organizations, amended the Act of 1938 by striking out the word "Armistice" and inserting in its place the word "Veterans." With the approval of this legislation (Public Law 380) on June 1, 1954, November 11th became a day to honor American veterans of all wars. . .

What happened to the part about being 'dedicated to world peace.'  Let's listen for that language next November.