Danny Amendola gives insight on how far off the Patriots were with their offer and provides an honest opinion on playing for Bill Belichick. (1:45)

BOSTON -- Wide receiver Danny Amendola detailed the factors that led him to leave the New England Patriots for the Miami Dolphins as a free agent this offseason, saying that after three years of taking a pay cut the team's offer fell significantly short compared to others he had received.

"I came in with an open mind. I understand Bill [Belichick] runs a tight ship, and he hasn't been known to pay his players, really. I understood that I gave money back to him so I could play for him and play for my teammates and fulfill my side of the contract, and at the end of the day, I had faith that he was going to give me an opportunity to stay," Amendola told ESPN.

"When free agency broke, I came to the realization that he wasn't going to really come close to any of the other offers I had," he said. "I had to make a decision for my family and go down to Miami and continue my career there."

Amendola, who signed a two-year, $12 million deal with the Dolphins that included $8.25 million in bonuses and guarantees, was in New England on Friday as the featured guest at the American Cancer Society's "Relay for Life" event at Merrimack College in North Andover, Massachusetts, and ESPN rode with him to the event.

Speaking to the crowd at the event, Amendola praised Belichick as the "best coach to ever coach the game." In his interview with ESPN, Amendola also discussed how it was challenging to play for Belichick at times.

"It's not easy, that's for sure. He's an a--h--- sometimes. There were a lot of things I didn't like about playing for him, but I must say, the things I didn't like were all in regards to getting the team better, and I respected him," he said. "I didn't like practicing in the snow, I didn't like practicing in the rain, but that was going to make us a better football team and that was going to make me a better football player. It wasn't easy, and he'd be the first to admit, at the [Super Bowl] ring ceremony, that it wasn't easy playing for him. The silver lining was that we were at the ring ceremony."

Part of the Patriots' success under Belichick in the salary-cap era has been maintaining financial discipline and building a strong middle class on the roster, and that business side of the game is something Amendola said he learned early in his career when he was cut by two different teams before playing in his first career game.

Meanwhile, Amendola said the Patriots' loss to the Eagles in Super Bowl LII still stings, and it hurt to see cornerback Malcolm Butler benched on defense for the game.

"I have my thoughts about it because I was out there putting my blood, sweat and tears out on the field that night, and one of our best players wasn't on the field," he said. "To tell you the truth, I don't know why. I did ask, but I didn't get any answers. I can't make decisions like that, so I don't necessarily worry about it, but I know Malcolm is a great player and he could have helped us win. For whatever reason, he wasn't out there. He's going to play more football in his career, and he's going to be a great player for a long time."

Asked if he sensed the Butler decision hurt his Patriots teammates, Amendola said, "Yeah, I did, honestly. Nobody really got an explanation for it. He's a brother of ours. He was a brother of ours that year. And I hate to see a guy who worked so hard throughout the season not get a chance to play in the biggest game of the year and really get no explanation for it. With that said, I don't know how the business aspect went into that decision. I don't know how the personal aspect went into that decision between him and Bill. But as a friend, I would have loved to see him on the field that day."

Saying he will always call Boston home, Amendola briefly seemed to get choked up when speaking of his friendship with Patriots receiver Julian Edelman while thanking him for raising his level of play.

Amendola, who referred to Patriots owner Robert Kraft as "a friend," also spoke with excitement about joining the Dolphins, saying he has spent time at the team's facility in advance of the voluntary offseason program so he can get to know some of his new teammates better. Amendola has already caught passes from quarterbacks Ryan Tannehill and Brock Osweiler, along with many of the team's skill-position players, at a different location.

Dolphins coach Adam Gase has talked about changing the culture in Miami, and signing the 32-year-old Amendola -- who said he envisions the window is closing on his career in the next three or four years -- has been part of that approach.

"I'm not the oldest guy on the team, but I've been around for a while, and I know what it takes to win a championship, I know what it takes to have a successful atmosphere," Amendola said. "I'm really excited just to share my knowledge in that respect, be a good teammate, whatever they ask me to do, and trying to catch as many balls as possible. ... It's a great opportunity to make new friendships and explore other football avenues. I'm really excited to continue to play."