Williams was a key offseason pickup from Atlanta as part of a trade that the Hawks made largely because they wanted to clear salary-cap space. His numbers sagged last season as he came back from an ACL tear he suffered in January 2013.
"I didn't think I was going to be the same player [after ACL tear]," Williams said. "I knew I wasn't going to have the foot speed I once had, I knew I wasn't going to be able to jump as high. But it taught me to be smart and I worked on different parts of my game. Being able to evolve and change my game to win this award is very gratifying."
Williams rebounded strongly with the Raptors, averaging a career-high 15.5 points in 80 games off the bench. He was particularly crucial to the team in December when star DeMar DeRozan went down for an extended period with an injury.
"I'm excited about Lou, a guy who comes back from an injury and works hard and is dedicated to getting his body right," Raptors coach Dwane Casey said Monday. "Nobody knew what he was going to bring to the table ... whether he was damaged goods or whatever. He's proved to everybody he's the old Lou Williams."
Thomas averaged 16.4 points in 67 combined games with Boston and the Phoenix Suns.
"[The voting is] not in my control," Thomas said. "Congrats to Lou Will. I can't say too much about it. All I'm going to say is numbers don't lie. But congrats to him. He deserves it."
Williams led or tied for the team lead in scoring 18 times off the bench, second most for a reserve in the league. In March he set a record for points in a quarter with 21 in the fourth period against the Cavaliers.
"[The award} is just a reminder than anything is possible if you say faithful to the grind and stay faithful to the things that put you into a position to be successful," Williams said.
Williams quickly became popular with Canadian fans for his laid-back demeanor and ability to score quickly. He achieved another level of celebrity when Raptors part-owner Drake referenced him in a recently released song called "6 Man."
"He finished a lot of games for us and helped us win a lot of games," DeRozan said. "He's a cool dude. Lou Will is the coolest dude in the world. If you can't tell on the court, now you know. He had a song about him and now he's got an award to go with the song."
Williams scored 10 points but was just 4-of-16 shooting in the Raptors' Game 1 overtime loss to the Wizards on Saturday. He will be presented with the trophy Monday afternoon at a ceremony in Toronto.
Thomas invigorated the Celtics' offense since his arrival at the February trade deadline and was a key element in Boston's late-season surge to the playoffs. He acknowledged that he might have had a better chance at winning the award if he had been with Boston longer.
"You could say that. It had a lot to do with winning, too," Thomas said. "Since I got to Boston, things changed, we started winning, we got into the playoffs, so you're definitely going to hear about more individual success than when I was in Phoenix.
"I thought I did a pretty good job in Phoenix as well. But it is what it is. I can't control it. I just go out there and play, and try to put myself in best position as possible."
Celtics coach Brad Stevens had previously lobbied for Thomas to win the award. Asked Monday about Williams earning the honor, Stevens said, "Lou Williams is a very deserving candidate, but Isaiah would have been my vote."
ESPNBoston.com's Chris Forsberg contributed to this report.