Well the Cayman Islands are a chain of islands, as opposed to just one island. There are 3 islands; Grand Cayman, Little Cayman and Cayman Brac each of which is unique in its own way. Grand cayman is the primary island, and is home to the territorial capitol of George Town...it is also highly built up in comparison to the other two islands. The West End of the island is home to the airport, and the majority of the hotels and resorts, the East End is primarily undeveloped and is mainly marshland. Little Cayman is in a much more natural state compared to Grand Cayman, it has significantly less lodging options but is also known for it's more natural beauty, and scuba diving. Lastly there is Cayman Brac, which is a bit more built up then Little Cayman but is still more natural than Grand Cayman. As a British territory, English is the most commonly spoken language on the islands...although the local dialect does differ from British or American English.
Anguilla is also a British territory, the common language is also english but a local variety of creole is also spoken. Anguilla is tourist island, but due to multiple Hurricanes over recent years, and the 2008 financial crisis it is struggling a bit more then some islands closer to the U.S. Still it is worth a look at, they need the business and it is a very "beautiful" place.
EDIT: The people will treat you with great respect in either location, tourism is the most important thing on these islands and they want your business.