Within walking distance from the Hyatt and just a short drive from The Boca Resort, Mizner Park is a great place to do some shopping or get a bite to eat. Here, you'll find bars, restaurants, coffee shops, chain stores (Lord & Taylor!) and boutiques. If there is rain (knock on wood!), check out the iPic movie theater. They have big comfy seats, blankets, and serve free popcorn. In the premium seats, they offer wait service for drinks and heavier bites.
Also within walking distance from The Hyatt, and a short drive from The Boca Resort, Royal Palm Place is another great plaza to check out. Here, you will find more bars and restaurants, and a few great spots for a casual breakfast, plus higher end boutiques, art galleries and salons. A touch "fancier" than Mizner Park, but a great place to get frozen yogurt and walk around.
There is certainly no shortage of things to do in Delray Beach, recently named "The Most Fun Small Town in America". Whether you're into the beach, shopping, dining, wine tastings, festivals, arts, or cultural experiences there is something for everyone. A short 10 minute drive from each hotel, this is a great place to spend the day at the beach (there's a beautiful boardwalk) or walk around town. There are tons of bars which make it great for a night out. Atlantic Avenue is the main strip. Off that, you will find Pineapple Grove Arts District. This is a more lively scene (day & night) than Mizner and Royal Palm. Strongly recommend!
If you're feeling ambitious, check out The Flagler Museum. When it was completed in 1902, the New York Herald proclaimed that Whitehall, Henry Flagler's Gilded Age estate in Palm Beach, was "more wonderful than any palace in Europe, grander and more magnificent than any other private dwelling in the world." Today, Whitehall is a National Historic Landmark and is open to the public as the Flagler Museum, featuring guided tours, changing exhibitions, and special programs.