Coming to medieval history as a complete novice, this question was a puzzle to me. The terms seem to be interchangeable, so why not stick to one? I thought you might wonder too, so here is the explanation.
The Middle Ages were so named by humanists in Italy in the late 15th century. They were working to revive Classical learning and culture, and looked back to the ancient Greek and Roman Empires as the last high point of culture. In between the two, was the 'Middle Ages'.
The word 'Medieval' is the adjective used for the Middle Ages. It comes from the Latin, meaning 'middle age'! So, Middle Ages and Medieval Period mean the same thing.
There is much debate over exactly when the Middle Ages began and ended. It covers roughly 1000 years, from the fall of Roman civilisation to the Renaissance. But Roman civilisation didn't collapse overnight, and the Renaissance came to different places at different times.
The web site medievalplus.com gives the following dates:
The Beginning of the Middle Ages
476 AD Regarded as the end of rule across Europe by the Roman Empire. Although Roman control of many parts of Europe had ceased several years previously due to rebellions and uprisings; in fact the Roman armies finally left Britain almost sixty years prior to that date.
The End of the Middle Ages
1453 The capture of Constantinople by the Turks
1453 The end of the Hundred Years' War between the English and the French
1492 The Muslims’ being ejected from Spain
1492 The discovery of America by Columbus
1517 The Protestant Reformation starting
Many people like to use round numbers to make it easier and say that the Middle Ages was 500AD to 1500AD. The second date is when the modern world is considered to have begun. Historians have also put forward cases for the Renaissance beginning in the 14th and even 13th centuries, and sometimes as late as the 17th century, but the 15th century is a good general date to use, and gives us that round number of 1000 years.
Now, just to complicate matters, the Middle Ages is split into three periods:
1. The Early Middle Ages (6th to 10th centuries)
2. The High Middle Ages (11th to 13th centuries)
3. The Late Middle Ages (14th to 15th centuries)
It was the first period which was known as the Dark Ages. This was originally named because it was assumed to be a time of ignorance and barbarity, and little was known about it. As historians learn more about this period, the term has fallen into disuse. I wrote about this in another blog post.