Stafford, the 37-year-old Los Angeles Rams signal-caller, received 24 of 50 first-place votes while the New England Patriots' Maye - who will play in Sunday's Super Bowl - received 23.
Buffalo Bills star Josh Allen placed third, while San Francisco 49ers running back Christian McCaffrey - the only non-quarterback on the final shortlist - finished fourth.
Stafford led the league this season with 4,707 passing yards and 46 touchdowns, but his Rams just missed out on a Super Bowl appearance with a 31-27 defeat to the Seattle Seahawks in last month's NFC Championship clash.
"This team was really special. It didn't end the way we wanted it to, but, man, I'll never forget it," said Stafford, flanked by his four young daughters.
"I'll see you guys next year," Stafford added.
Also at the NFL Honors ceremony in San Francisco hosted by "Mad Men" star Jon Hamm, the Patriots' Mike Vrabel was named head coach of the year for a second time.
In his debut season as New England coach, former Patriots player Vrabel took a franchise that had won just four games in each of its two previous seasons all the way to this Sunday's Super Bowl.
Vrabel was previously the 2021 award winner with the Tennessee Titans.
Jaxon Smith-Njigba, the Seattle Seahawks wide receiver who will take on Vrabel's Patriots at Super Bowl LX, won offensive player of the year.
His league-high 1,793 receiving yards was the eighth-most in NFL history.
Cleveland Browns star Myles Garrett was named the defensive player of the year after setting the NFL record for sacks in a single season, with 23.
NFL legends including Drew Brees were named to the Pro Football Hall of Fame.
But, as had already been widely leaked, six-time Super Bowl-winning former Patriots head coach Bill Belichick failed to be elected to the Hall in his first appearance on the ballot.
Belichick, 73, is widely regarded as one of the greatest coaches in the history of the NFL.
But his career was also tinged with controversy, notably the 2007 "Spygate" scandal, when he was fined a record $500,000 after the Patriots were caught illegally filming signals by opposing coaches during games.
