Doing a little research online revealed that the two days are very much the same, simply their roots are set in different era's – one began due to observance and memorial of those who made great sacrifices in the civil war and the second was an amalgamation of observance and memorial of World Wars I and II.
Per the Us Department of Veterans Affairs website:
History of Veterans Day
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.
Thus began Armistice day...
Memorial Day began as Decoration day and over time grew to become the holiday we know it as, and in 1938 Armistice day became an official legal US holiday. This is a great opportunity for your older elementary aged student to cover history of the US, and have their own comparisons and discoveries as to how the holidays are similar but different.
Post World War II it became known as Veteran's Day – and a great activity would be for your students to trace the timeline of what twists and turns have happened over time and with laws to make the holiday we now know it to be.
Good luck and come back to share with us what you have come up with!