Kaihautū of Te Papa Dr Arapata Hakiwai said it was an honour to return these taonga permanently to the people of Hawai'i.
"These priceless treasures have so much to tell us about our shared Pacific history. We are honoured to be able to return them home, to reconnect them with their land and their people.
"Woven into these taonga is the story of our Pacific history, with all its beauty, challenges and complexity."
Hakiwai said the treasures reminded him of the proverb He Toi Whakairo, He Mana Tangata, meaning where there is artistic excellence, there is human dignity.
For native Hawaiians, all featherwork was reserved exclusively for the use of their ali'I or royalty.
The feathers of 20,000 birds were needed to make the cloak.
Skilled trappers caught the birds by snaring their prey with nets, or using decoy birds to lure them onto branches coated with a sticky substance.
In 1779 Kalani'ōpu'u greeted Cook after his ship made port in Kealakekua Bay.
As a demonstration of his goodwill, Kalani'ōpu'u gifted the feathered cloak and helmet he was wearing to Cook.
Cook left Kealakekua Bay in early February but unexpectedly returned a few days after his ships were damaged in a storm.
Following a series of disagreements with the Hawaiians, a confrontation ensued and several Hawaiian chiefs and a large number of commoners were killed or wounded.
Cook and four of his men also lost their lives.
The garments passed through the hands of various museum owners and collectors after they were taken to England.
Eventually they came into the care of Lord St Oswald, who unexpectedly presented his entire collection in 1912 to the Dominion Museum in New Zealand, the predecessor of Te Papa.
Bishop Museum president and CEO Melanie Y. Ide said they were humbled to accept the responsibility of caring for the garments.
"With their extraordinary presence, they give the people of Hawai'i a tangible connection to the past, and to ancestors whose mana remain strongly rooted.
"The impact of this gift will be felt for generations, and we will honour Te Papa's inspirational act of leadership and generosity with our commitment to strengthen the kinship between our peoples and institutions."