The 28-year-old Diaz had 11 assists and 12 penalty minutes in 46 games for the Canadiens this season.
The native of Baar, Switzerland, has 41 points (four goals, 37 assists) and 48 penalty minutes in 128 NHL career games but was a healthy scratch of late with Montreal.
"This is the business. This is the way it is," Diaz said Monday on a conference call. "As a player you want to play as much as you can and you want to help you team."
Diaz joins a Canucks' blue-line that has suffered a number of injuries in recent weeks, with both Christopher Tanev and Kevin Bieksa currently out of the lineup.
"As a sportsman, as a hockey player, you respect everything (teams) do," he said. "(The Canadiens) always treated me good. They helped me learn a lot things to play in the NHL.
"I'm really excited for the new opportunity."
Meanwhile, Weise never seemed to fit into the plans of Canucks head coach John Tortorella. The Winnipeg native had three goals, nine assists and 42 penalty minutes in 44 games this season, mostly while playing on the fourth line.
"I think I can bring a physical presence to the (Canadiens)," said Weise. "I'm a good skater and I work hard and my success is down low and hanging onto pucks and being physical and trying to get to the hard areas to score goals.
"I'm not sure where I'm going to fit in just yet. I'm just looking forward to coming in and trying to play hard and trying to find a spot."
In 162 career NHL games, the 25-year-old has 10 goals, 16 assists and 185 penalty minutes. Weise's name had been mentioned in trade speculation in recent weeks, but Monday's deal still caught him off guard.
"(The trade) comes as a little bit of a shock to me," said Weise. "I think any time you hear rumours around you think about it but there's so many (rumours) that go around and you just never know.
"I didn't put too much thought into it ... I was shocked when I heard about it today."
Weise was expected to arrive in Montreal on Monday night and practise with the Canadiens on Tuesday morning, while Diaz was set to meet up with his new teammates the same day in Boston.
Both Weise and Diaz will get a chance to face their old teams this week when the Canadiens host the Canucks on Thursday night at the Bell Centre.