I always have to qualify what you are looking for to give you a proper answer. Cancun honeymoons, Riviera Maya honeymoons and Cozumel honeymoons give you both the history and flavor of Mexico along with beautiful beaches lining the Caribbean Sea.
The Caribbean and Mexico are very diverse, so you need to do a little research on the different destinations. Jamaica honeymoons offer what most couples imagine with the Caribbean, white sand beaches, turquoise blue water, rolling green hills, rivers, waterfalls, reggae and rum. Punta Cana honeymoons have a big Spanish influence so a nightly show is more likely to be salsa and merengue instead of Bob Marley.
A Cancun honeymoon gives you very modern luxury resorts, shopping and tons of nightlife, where a Cozumel honeymoon is low key casual. I like to say it's similar to Miami Beach vs Key West or close to the same thing.
For those of you looking for the road less traveled, the Caribbean offers St Lucia honeymoons and Antigua honeymoons. Mexico offers Zihuatanejo honeymoons and Huatulco honeymoons for the explorers. Both destinations are on the Pacific coast.