Hinckley, leader of the 12 million-member Church of Jesus Christ of Latter-day Saints, conducted the 90-minute service for Maxwell, who died July 21 at age 78.
About 4,000 people attended the funeral, either in the LDS Tabernacle on Temple Square or watching on a screen in overflow seating in the Assembly Hall and Joseph Smith Memorial Building. Many local dignitaries were in the audience, including U.S. Sens. Orrin Hatch and Bob Bennett, and Congressmen Chris Cannon and Jim Matheson, his mother, Norma Matheson, and Monsignor Terrence Fitzgerald of the Catholic Diocese of Salt Lake City.
All of the apostles were there, including David B. Haight, who is nearly 98 years old and had to be helped to the platform. The rose-colored plush seat where Maxwell sat for 21 years was symbolically vacant.
The Mormon Tabernacle Choir, dressed in black and white, performed several emotion-packed hymns, including "O Divine Redeemer," "Come Unto Him" and "Be Still, My Soul."
The service featured five speakers, four Mormon leaders and Cory Maxwell, the apostle's only son.
Every speaker mentioned Maxwell's prodigious and sophisticated writings, his literary techniques, his kindness and humility, and the love of his family. Each described the popular speaker's faith in the continuation of life into the eternities.
"When he opened his mouth, we all listened," Hinckley said. "Each talk was a masterpiece, each book a work of art."
Thomas S. Monson, first counselor in the church's governing First Presidency, called Maxwell "a giant among men."
"The theme of Maxwell's life could be summarized in two words: I serve," Monson said. "Neal Maxwell served his country, church, family, fellow man and his God. He increased in wisdom, stature and favor with the Lord."
During Monson's last visit with Maxwell, the dying apostle asked: "Tom, do you think Brother Harold B. Lee needs me on the leadership committee where he is and where I shall be soon?"
Lee, 11th president of the LDS Church, died unexpectedly in 1972 at 73.
Cory Maxwell described his father as a family man who took his children and grandchildren on vacations, most recently a bus trip through Mormon history sites; organized grandchildren firesides where he would discuss church topics; and served as a kind of master teacher to them all.
A week ago, Maxwell called all of his 24 grandchildren to his bedside and spoke to them as "Father-Adam," expressing his love for them. He then gave each a personal blessing.
"He taught us how to die," Cory Maxwell said.
James E. Faust, second counselor in the First Presidency, shared Maxwell's humble Salt Lake City origins.
"Although not in the same class, we were both Granite High farmers," Faust said. "I milked cows; he raised pigs."
Faust recalled that their English teacher once gave Maxwell a low grade on an assignment. On it she had written: "Neal, you can do better than this."
The fact that she believed in his talent may have launched his lifelong interest in writing and thinking, Faust said.
Boyd K. Packer, acting president of the Quorum of Twelve Apostles, is known for his efforts to make sure Mormon funerals are appropriately solemn and serious. Still, he couldn't resist a humorous anecdote.
The last time Maxwell attended the apostles' weekly meeting in the Salt Lake Temple, he was weakened by the cancer, leaning heavily on a cane. When he spied Haight, hobbling in with his cane, Maxwell lifted his own and said, "En garde!"
Haight rose to the challenge "and a kind of fencing match ensued," Packer said, quickly assuring the assembled Mormon faithful that "the match broke off without bloodshed."
Then he launched into a discussion of the church's teachings about the hereafter. Drawing on LDS scriptures, Packer said that Mormons believe that while the body decays in the ground, the spirit is born again in the presence of God and couples are reunited.
He read from an essay titled "On Dying" that Maxwell wrote just weeks ago.
"Death is not an exclamation point," Maxwell wrote. "It's a comma."
But it is painful, said Hinckley, who lost Marjorie, his wife of 67 years, on April 6.
"At funerals we speak platitudes intended to give comfort, but only those who have walked through this dark valley know its utter desolation," said Hinckley, 94. "To lose one's much-loved partner with whom one has walked long through sunshine and shadow is absolutely devastating. There is a consuming loneliness which increases in intensity and painfully gnaws at one's very soul."
Then comes "in the quiet of the night a silent whisper that 'all is well, all is well,' he said. "That voice from out of the unknown brings peace and certainty and unwavering assurance that death is not the end, that life goes on with work to do and victories to be won."
Hinckley assured Maxwell's widow, Colleen, that "the sunlight of faith will shine again and warm the fires of love within and about you."
He ended by quoting Shakespeare's "Hamlet": "Good night, sweet prince, and flights of angels sing thee to thy rest."
Maxwell was interred on the north edge of Salt Lake City cemetery, where the grave was dedicated by his son-in-law, Mark L. Anderson.