They are synonymous for the most part. However, plight is more intense in flavor and suggests that there is danger involved. A predicament is used for just an unfortunate or unpleasant situation. However, you could interchange the two and still be right almost all the time.

These are both synonyms of each other. Both are basically problems, oftentimes especially difficult, embarrassing, or even dangerous. Another good word meaning the same thing is quandary!