So here's the difference in a nutshell: Memorial Day is time to remember those service people who have died in the line of duty, while Veterans Day is a time to honor all service people, alive and dead, of their contributions to maintaining and securing our freedom.
I did a little more digging to find out a little more about the history of both. So here are a couple of tidbits on when both were started. The first Memorial Day started back around 1865, but was originally known as Decoration Day. It began as a way to honor the soldiers, both Confederate and Union, that fell during the Civil War. As time has worn on and our country has participated in more wars, the commemoration of fallen soldiers has extended to all fallen Armed Forces service people.
The first Veterans Day actually took place in 1919, but was known as Armistice Day. It was a day to remember the heroism of those who died during The Great War. Roughly seven years later, Congress requested that it become an official national holiday, which was passed. Again, as time continued and another World War commenced, the commemoration was extended to all soldiers regardless of the war they served in and whether or not they had fallen, and thus, it was renamed Veteran's Day.
So there you go. A little history and a small comparison of two similar but rather different holidays. I hope that was more interesting than just a regular classroom history lesson. Short and sweet.