Quick Answer: Technically, boogie boards and bodyboards are the same things. 
​

The main difference is that a bodyboard is the true name for the board and sport while the name "boogie board" came from a business called Morey Boogie, founded by Tom Morey.

Just as a person might mistakenly refer to a hot tub as a Jacuzzi or a tissue as a Kleenex, someone might mistakenly refer to a bodyboard not associated with the boogie board brand as a "boogie boards." That is because Jacuzzi and Kleenex are companies that makes those products, not the actual product name itself.

The original boogie boards came about in 1971 when Tom Morey created it to ride the surf out in Hawaii. Morey was a major fan of boogie-woogie jazz music, which is where the Morey Boogie name came from. Tom build that bodyboard business, Morey Boogie, to one of the largest producing board companies at that time and one that had the biggest impact on the sport in years to come. He later sold his business to a toy company, which created the boogie board name.

As other companies began manufacturing similar boards, they were not allowed to use the trademarked boogie board name, which is why they began referring to them as bodyboards.

So yes, there is truly no difference between boogie boards and bodyboards.