What was the difference between an Angle, a Jute, and a Saxon? And why have people in the last few centuries cared so much about their affiliations with these ancient tribes?

This answer is going to be a relatively simplified version and if anyone would like to correct something or expand upon what I've said, please do.

The Angles, Jutes and Saxons were initially different tribes from continental Europe. The Angles came from an area in modern-day Germany called Angeln on the Jutland Peninsula. They would go on to settle a number of parts of England, for example East Anglia. Link

The Jutes, on the other hand, came from the northern part of the Jutland Peninsula and mainly settled in Kent. Finally, the Saxons appear to have come from Holstein, just south of the Jutland Peninsula, the areas along the coast of modern-day Germany and the Netherlands as well as further inlands. They settled in the south of England, creating the kingdoms Wessex, Sussex, Essex and Middlesex. Link #2

Over the centuries these distinctions disappeared, which you can read more about here. Alfred the Great would, during the latter half of the 9th century, go on to call himself the first King of the Anglo-Saxons. His grandson, Æthelstan, would in 927 become the first King of England, having conquered the "Viking kingdom" of York, which had originally been invaded by Danish Vikings in 866. That, on the other hand, is an entirely different question.

So to sum up, the Angles, Jutes and Saxons all came from different areas of the continent and settled in different parts of modern-day England. Eventually, the distinction between them disappeared and by the 10th century they were united under one king.

The acronym WASP means White Anglo-Saxon Protestant. The Anglo-Saxon part refers to their English ancestry, which is a far cry from reality, given how the English people of today, stretching back centuries, are a mix of Britons, Anglo-Saxons, Scandinavians, Normans and so on. It does not specifically refer to the Anglo-Saxons.