Both areas are excellent Spring and Fall.
My choice would be based on your goals. You will need to rent a car for both locations.
If you are bringing family and want to tour historical landmarks etc (Hatteras)
Ease of getting there (SPI - Houston-Brownsville-car 35 minutes)
Both locations have beach houses to rent for extended stays.
many kiters take advantage of SPI without a car rental. it can be done. (not that i recommend it, but it is not only completely possible, its not really that difficult) : free/cheap airport shuttle to island, free island shuttle service, and $6 taxi rides to almost anywhere around the island. flat water and wave riding within walking distance of your hotel/condo, assuming you pick the right location.
you dont really get to pick which airport you fly into around here, it just has to do with what airline and the schedule etc. its either BRO or HRL, basically everyone will connect through HOU or DFW. BRO is 15min closer to island kite paradise, HRL has better shuttle and car rental services on site.