A rival is always somebody (or an entity) you compete with, for being comparable. So for example, Coca-Cola and Pepsi can be considered rivals, because they are both similar in size and compete for business; you would not consider a cheap alternative of these brands a rival of them however, even though they compete for business, for they are not on the same level. Hence, "rivalry" is rather a personal competition, as pointed out above. An opponent however is one (or others) who opposes you. There can be a rivalry with an opponent, but this is not necessary to render them an opponent. So for example, two professional boxers in a ring - each would regard the other an opponent, for they are about to fight one another. They may even have a rivalry, too. Quite differently, you may find an opponent in your parents, when they express disapproval to one of your ideas or plans - they are not your rivals, but they are an opponent to one of your ideas. I hope the above makes it clear, but let me know if it does not!

The Cambridge Dictionary refers to 'rival' as a person or thing competing with another for the same objective or for superiority in the same field of activity. The same dictionary refers to opponent as: someone who competes with or opposes another in a contest, game, or argument. So, you would not find an opponent at work, but a rival. However, in a tennis match, you would compete against your opponent or your rival, but they would have slightly different nuances. There are added meanings for both words, but that is what you asked....

An opponent is a competitor - someone who plays against you in tennis, for example. A rival is more personal - some who you regularly compete against and dislike. Similar to enemy, but not as serious.

Thanks Ben.

Moved this to an answer!