You got some really nice beaches right there, on Maui. It's a little touristy and not that easy to find that private corner on a beach but definitely beautiful.
Making a decision between Kauai and Big Island is like deciding between oranges and apples. Too bad that you did not mention how long your overallstay in Hawaii is. If you take a trip to one of these 2 islands, you want to stay 3 nights on that island.
What are major differences? Kauai is the Garden Island, meaning it's green and lush because it has frequent rain showers but they don't last long. You can walk on stretches of white sand beaches for ever, at lot of the times even without meeting any people! Meaning no Maitai or restaurant either.
My suggestion is, do some island hopping to the Big Island (interisland flights are decent these days). Even if you are not into activities, at least you have the option to choose from many different ones, many beyond the ordinary...Stargazing on Mauna Kea Summit, Visiting Waipio Valley and Pololu Valley Lookouts (even the walk into the valley is easy), attend one of the Hawaiian luaus (I like the one at the Marriott, Waikoloa Beach Resort) or even include a day trip to Hawaii Volcanoes National Park (nothing compares to seeing a live volcano!).
Now, you only need to decide where to stay on the Big Island. The West coast, the Kohala Coast, is the sunny side with lots of white sand beaches (most famous Hapuna und Mauna Kea Beach). Hotels on this coast can be expensive. If you are on a budget, I recommend a vacation rental with a short drive to the beaches.
Hope this helps planning your trip. Have a wonderful stay in Paradise. Aloha, Pua