The words have similar meanings, but are not entirely interchangeable. The simplest difference that feels right is that 'contested' is usually to do with things that are on some level factual, whilst 'controversial' is usually to do with things that are more emotional. A 'controversial' topic is usually one where people's opinions are guided a reasonably large amount by emotion or morality. For example, American politics right now is a quite controversial topic: some people feel very strongly about each of the different candidates. On the other hand, a contested island is (in theory) a matter of principles. It should be the case (although of course it's not really...) that the country to which an island belongs is a matter of rational debate. It's also something that the average person is probably less likely to have a strong opinion of.

Here's another way to look at them. * controversial - hotly debated with a lot of different arguments being professed. Think of "contro" as "opposing", and "versial" as "voiced" (as in "converse"). It means "argued about to a great extent". "controversial topic" means there are a lot of contentions surrounding the topic. * contested - being vied for by different parties with concrete interest at stake. "contested island" means several parties are fighting for the control of the island. Think of the "contests" in which many contestants compete for a prize.