Just adding to your list, since I live there.
What can you do on the BI that you can't on Oahu?
See the big telescopes.
See black sand and green sand beaches.
Drive up a mountain along fragrant redwood trees.
See cows placidly grazing in paradise.
Be alone on a nice beach.
Enjoy volcanically heated natural pools.
See where Kona coffee is actually grown.
And, finally, and most important to me. . .
Go on Hawaiin time. . . Just let the peace sink in. . .
*Drive from the beach to 14k elevation, to the top of Mauna Kea, while still wearing your swimming clothes, to be higher then all of the clouds, the closest to heaven and find peace and tranquility like you've experienced before. And yes... It does snow there at times.
*Drive to Ka Lae in Kau, which is the southern most point of the Big Idland, State of Hawaii and all of the 50 US states. And once there, observe the eerie currents of the ocean that direct the waters, allowing a still and most times, motionless calm between them.
*Visit Waipio valley the largest windward valley on the windward side of the island. After all, it's been filmed in many movies so you know it's worth seeing.
*Hike into a volcano crater, Kilauea Iki and listen and see some of our indigenous birds of our island, soon to be extinct.
*Swim with Manta rays in Keauhou.
* Visit Pahoa town, the town that was almost taken by Pele (Our Fire Goddess, Deity), "lava" in 2014/2015. No exaggeration here... The governor declared an emergency proclamation, because Pele, "the lava flow", if continued on its track would have separated the town from the rest of the island, causing what was about to be, the island within an island. Fortunately, the lava stopped. But only after it took 1 home, created some amazing landscaping to the transfer station, it took a gravesite, and stopped a few hundred feet from a home and the main road leading into the town. The Ariel view of the flow shows the destruction and miracle of this little town. And.... The hardened lava "pahoehoe" that you would see is literally less then 2 years old. Talk about a brand new new land.
*If on the Big Island, you canexperience the tourist trap called Waikoloa, head up to Kohala mountains and see the island of Maui on a clear day, with the most beautiful breeze and lush acres of farmland, head down to Waimea and be amazed with how the land is divided by a dry desert like environment on one side then lush greenery on the other side. Waimea is our Paniolo "Cowboy" country town where the air is nippy, chilly and very country inspired, then drive along the Hamakua coast and experience the 3 horseshoes that lead to Akaka Falls then into Hilo. Once in Hilo, visit the Tsunami visitor center and see/listen to the stories of the 1946 April fools day Tsunami that devastated that very town. From Hilo you could drive through the Panaewa rain forest to Pahoa, then down to Kalapana and see how that little town has grown since it was taken from Pele in 1990. From Kalapana head out of Pahoa, up to Volcano and experience the coldest temperatures on the island, not including Mauna Kea, and beautiful forests that surround our active volcano. After leaving Volcano head down to Ka'u to see Punalu'u (The black sand beach) then to Papakolea (The green sand beach) prepare to walk a few miles to get there, then drive to Captain Cook and visit Pu'uhonua (The City of refuge), then head into Kailua Kona then Back to Waikoloa if you're staying there.
** I highly suggest that you never ever take any lava rocks from the island back home with you. But you can do your own research.
* A lot to see and experience. And if you're gifted with a 6th sense... intuitive, a medium, psychic etc... or have any other spiritual gifts, you're in for a much deeper and richer experience.