I have been to both in the summer and each has their own unique feel. Where to go is all dependent on what you like to do.
Myrtle Beach
Myrtle Beach is very much so an active place. There's an amusement park right off the beach, the Boardwalk has many gift type shops and bars. There's live action plays, minor league baseball and even a college with a D1 football team close by. Broadway at the Beach has many family attractions like The Broadway Grand Prix (8 track Go-Kart Park), Ripley's Aquarium and Wonderworks. They have a Hard Rock Cafe there and many large chain stores (Ron Jon Surf Shop, etc). They also have/had some fun little shops like the Retro Store (70's-90's memorabilia) and the Stupid Factory (sarcastic and funny graphic tees, socks, coffee cups and the like). The Boardwalk can get packed with partiers on Friday and Saturday nights. There are several mini golf and water parks in the area. The food scene is decent, but not up to various other summer beach towns. There are some rougher parts in the area, especially south of the Boardwalk where there are several abandoned resorts right off the beach. The resorts are very tall and right up next to the beach. The beach north of the Boardwalk (we stayed at HGVC Ocean 22) was kind of meh. Getting into the water was not as fun as there's a drop off where shells collect and it can get a little uncomfortable to step through. If you are into golf, there are plenty of courses to play, many at reduced rates (or free) with your stay and many resorts.
Hilton Head
Hilton Head is more laid back and upscale. The community is very organic (take pride in local businesses). So many outdoors type of activities. As mentioned above, there are miles and miles of bike trails. Many resorts have deals with bike rental places and deliver your bikes to you. There's hiking, kayaking, fishing, and even bird watching. The beach is FANTASTIC! One of my favorite beaches and rivals some of beaches on Michigan's Sunset Coast. It has such a small decline and is easy for anybody to get in and out of. The resorts are pretty short (tallest is something like 6 stories) and they are set back from the beach so you just enjoy them. There are commercial gift type shops, but most are locally owned and have decent stock. You are close to Savannah Georgia and that's always a great trip. Then there's the food. One of the top foodie places in the US, you can get everything from diner food to 5 star restaurants. There's a thread in the Marriott forum just on the restaurants and its 30 some pages long. I discovered my love for Poke Bowls here. Had one of the best burgers I have ever had at The Frosty Frog in Coligny Park (the main commercial center right off of Coligny Beach). The golf is great here too. There isn't as much for younger families not interested in outdoorsy type of activities. No amusement parks, boardwalks, water parks, mini golf and the like. At least we didn't see any. I believe the main drawback is everything is so spread out. To get to many of the restaurants, it's a decent drive. It might take you 10-15 minutes to get to some places if you are staying right off the beach.
Preference
My family and I like HHI much better than MB, but would go back to MB in heartbeat. We liked both very much. With older children (13, 19 and 23) HHI is a better choice for us, but they all liked MB too.