I have been to both places and can tell you plenty. This will all depend on what you like to do and what kind of feel you like ina vaction destinatioin. I will take each place one at a time.
Hilton Head-This is of course old low country South Carolina. Anyone that has been to the coastal areas of South Carolina or Georgia can tell you that this coast line is easily one of th emost romantic and historic places in the country. Savannah Georgia is perhaps tops on my list for romantic cities. It is perfect for a anniversary or romantic getaway. The whole coastal area of Georgia and South Carolina is charecterized by huge live oak trees with branches that sweep to the ground and have wisps of spanish moss attached. At once you can feel the sensation of stepping into history, romance and even a little ghost story. The places in this area are unlike any I have seen elsewhere. As you go out onto the island you immediately realize that this whole place is dedicated to resort living and everything, I mean EVERYTHING is planned out as far as infrastructure. The roads are even patterned, in some spots, after european circular throughfares. You know the type that you see in movies where an intersection is not a four way but a very clustered up circle where you drive around until you hit the road you need and then shoot to the right. Even the
Walmart had to break it's own corporate building and safety requirement regarding parking lot lighting to comply with the very strict building and development codes. GOLF I say that again GOLF is the main thing here. If anyone in your party loves golf this is heaven. There is another course every three feet or so. There is also lots of shopping. You can shop for small trinkets here or extremely expensive items of all kinds. There are many many dining options. Ah to the beach. This is where I place some great degree of importance. The beach here is okay. Not bad, not great, but okay. Lots of sharks. I do not care what anyone will tell you, I have fished these waters in SC, Ga., and North FLa for years and the sharks are abundant. Not to worry attacks are almost unheard of. I say this as you may wish to fish on the beach and you will pull in about 4 bonnet head sharks in the 1 to 1 1/2 foot range to every other fish out there you catch. The deep sea fishing on charters is great. The beach is great for layoing around and tanning. It is flat and has dark sand. It is a great deal like a darker sometimes muddier version of Daytona Beach if you have ever been there. Did I mention expense?? I think I did. Did I say VERY VERY expensive...or at least it can be. It has a high end resort clientel for the most part and caters to the Golf country club crowd. A round of golf at the Westin (excellent place) was well over $100 per person about 5 to 6 years ago. It can also be very crowded and hard to get tee times and seats at restaurants.
Gulf Shores-It is hard to beat the Gulf of Mexico for beaches and water. In fact Hilton Head cannot compete here. Absolutely hands down the gulf beaches are the best beaches you will find in the sooutheast as far as I am concerned. It doesn't really seem to matter where on the Gulf you go it is going to be gorgeous. White sand and untouched beaches all along the coastline. There is or at least was (5 years ago) little development here compared to nearby areas in Florida and certainly compared to Hilton Head. The ara has a beach and coast feel but maintains a special rural charm somehow. If you leave the beach and go north it looks like farmland and pine trees. There is a little shopping and a few good restaurants here. There is some great seafood here. There is not a great deal of anything in particular here but enough to make you happy and to have a good time. Development may have gone up though. I loved it. I was very relaxing here and very laid back. Jimmy Buffett would go here instead of Hilton Head...his sister owns a restaurant and lives here anyway.
I also loved Hilton Head for very different reasons. And now we get to the true cut in your decision. This is just my opinion but...
If you are taking the kids, the beach and pool is where you will spend the time. I would suggest Gulf Shores. You will have a better beach and less crowd with a water park or two as a diversion at Gulf Shores. There is outlet shopping and the fishing is also excellent offshore. Beach fishing is the pits. Offshore however is Phenominal! Golf is okay.
If you are going with DH or sig. other and romance is the key, go to Hilton Head. If golf is important definitely go to Hilton Head. If you like history, especially the romantic swashbuckling pirate type history, go to Hilton Head. If shopping and numerous great restaurants are enough of a diversion to where you will spend some time on the beach and a good deal of time doing other things=Hilton Head.
Have fun choosing. 
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