This is not the first instance where some random, often strange or inappropriate social media status was shared by a celebrity (be it entertainment or political) that may stir up a commotion or problems for that person. Often times, that same person will go on to make an announcement that this message was not legitimate, that their account was "hacked" or they were simply joking around because of the backlash they receive due to the message, and all-in-all it creates more publicity for them even if it is negative.
Tweet threats are not serious between celebrities. This is because they have their reputations on the line and they make too much money to risk having an actual screw-up that would change their lives. A lot of celebrities simply want to keep the limelight on themselves so they will say outlandish things to get attention.
Although those who live their lives in the public eye are no doubt highly aware that everything they do is subject to both negative and positive publicity, this was probably a genuine act that wasn't very well thought out. Justin Beiber irritates a lot of people, and Big Sean is human, after all.
Whoever broke this story is responsible for it, not Big Sean. He has denied sending the tweet and is in fact friends with Bieber. Big Sean is doesn't have to worry about competition from Justin Beiber just because he sang a song with Ariana. They were onstage in front of thousands of people. It's not like they were up to anything.