Just a quick few reasons:
- Being able to show in a lot more classes
- Worth more (though some reg. Horses do go to auctions/kill) if you decide to sell
- Known pedigree on paper, and breeder info, so if anyone owns him later on, they will know more about him.
If you sold him later on, and couldnt contact the owners, you would have mroe luck Googling his registered name to look for him (even maybe in show results etc) rather than googling "BillyJoBobbyMo" which could be a common barn/stable name.
If my horse/s (currently don't own any) could be registered, then I would register them.