Griffin, the Heisman trophy winner in 1974 and 1975, believes it is more difficult to win the award in back-to-back years than it was 40 years ago. For him, the off-the-field distractions were more stressful than any defense.
"I'll never forget that I was trying to do everything for everybody and (coach) Woody (Hayes) called me into his office," Griffin said. "He told me, 'You know what? It's going to make you soft. You can't do everything for everybody.'"
But unlike Griffin, who admitted it was a goal, Winston said he hasn't even thought about a second Heisman. Winston said he is more concerned with a second national championship.
"I'm going to be great, even better than I was last year," Winston said. "I'm going to continue to be myself."
Coach Jimbo Fisher said he and Winston have talked about the award, but not about winning another.
"I do (talk to him) from the point of the responsibility that goes with it with your character and the things you do," Fisher said. "Not about winning another one. If he just goes and plays well, that will take care of itself.
"He never thought about winning a Heisman going into last year. You can't worry about that."
There will be plenty who will.
The demands and expectations are higher than when Griffin made history. Winston is the leading candidate heading into the season with the defending national champion Seminoles likely beginning the year ranked No. 1.
Cleveland Browns quarterback Johnny Manziel, the 2012 Heisman winner, said that was the most difficult part for him in 2013.
"There is a lot of pressure," Manziel said. "You're the one that's on TV every week. You're the one who at the beginning of the year is already at the top of everybody's Heisman list. ... It's everywhere because it's the biggest trophy in college football.
"For me, I never really let it get to me too much, but at the same time, it was always around and it was always lingering no matter what went on throughout the season."
The Seminoles and Winston have tried to manage his off-the-field commitments.
Winston made few public appearances during the offseason. He was the closer on the baseball team, finishing with a team-best 1.08 ERA, and accepted a handful of football awards. Winston was honored in his hometown of Bessemer, Alabama, and his Hueytown High School jersey was retired. There was also a trip to the ESPYs.
The Florida State signal-caller said he has learned to be more guarded, but still loves having "all eyes on me."
Griffin said that isn't necessarily a good thing.
"They're in a fishbowl," Griffin said of Heisman contenders. "I mean, anything they do, Johnny, every move he made it was talked about. Jameis, same thing. They've got to be extremely careful how they handle themselves because whatever they do, people are going to know about it."
Winston is no stranger to unwanted attention.
He was suspended for three baseball games and completed 20 hours of community service after admitting he stole $32 worth of crab legs from a local grocery store in April. He faced criticism nationwide and was the subject of taunts and jokes in print, online and on social media.
Winston was able to handle what Fisher described as distractions during the national championship run. Winston was investigated for an alleged sexual assault, but wasn't charged by the attorney general.
The QB will also have a few on-field challenges.
The Seminoles lost two of their top three receivers and their top two running backs to the NFL. Those four accounted for 52.6 percent of the offense and 44 touchdowns.
Nonetheless, because of Winston, the Seminoles will be one of the teams to beat.
Despite his talent, Winston has continued to downplay questions about the NFL. He insists that playing college baseball remains a priority. It would be a challenge to prepare for the 2015 NFL draft while playing for Florida State's baseball team.
Once Winston does get to the NFL, former Seminoles quarterback and current Buffalo Bills starter EJ Manuel said his anticipation on his throws will be his most valuable trait in the league.
"You can't take away some of the throws he's making out there," Manuel said. "They were tremendous. That's why he earned the Heisman.
"I think, moving forward, his anticipation skills, obviously, is a sign of intelligence. So that's going to help him build toward his last couple years at Florida State and then move on to the NFL."
There was an atmosphere of curiosity around Winston last season after being the No. 1 high school quarterback in the nation. The hype is different now because of his dominance on the field and the criticism away from it.
"I don't have time to focus on bad things," Winston said. "I'm always looking forward, keeping a smile on my face and focusing on good things."
AP sports writer Tom Withers in Cleveland contributed to this report.