Why does Jose have it in for Conte so much? It's almost as though he sees Chelsea as his team and he can't stand somebody else managing them.
This "beef" goes farther back than just this season. Last season Mourinho was the first (that I know of) to credit Chelsea's League table lead to Chelsea playing "solid defense, and counter-attacking really well" (paraphrasing). Conte was told about it by the press (in a way to make it sound like a slight at Conte), and Conte took the bait, so the war of words began there.
Unprovoked, (start of this season) Conte said he hoped to avoid a "Mourinho season." That's a dammed thing thing to say (a "Mourinho season" is when his team heavily under-performs and he gets sacked, not when he's winning one of his 25 career trophies, huh?). That escalated the war, and Mourinho fired back with "I'm not going to lose my hair over Conte's comments." Conte's had hair transplant surgery, allegedly.
During Pogba's injury Mourinho said he doesn't whine about player injuries the way others do, and Conte took it as a shot at him. So he pissed himself at the next presser (10:52):
Mourinho's passion was recently questioned by some media and fans because he doesn't run around on the sidelines anymore, so he clarified why he isn't as animated as he used to be, and that his "not acting like a clown" shouldn't be misconstrued as lack of passion.
Conte either thought he was talking about him, or used the opportunity to take another shot at Mourinho. So Mourinho fired back with the "at least I've never been suspended for match fixing."
I'm convinced you don't actually watch unedited press conferences and instead just read partial quotes in the papers or online articles, otherwise you'd know that the media does this to
every manager to try to get the next big headline. It's just that some managers lack the sense required to avoid falling for their schemes (in the video above, Chelsea's media manager finally cuts Conte off and tells him, "Dude, he didn't even say your name. The press is just egging you on and you're falling for it. Let's move on.").
Edit: keep watching the video above past his initial response. When the translator starts translating, Conte realizes he might have been had and tries to clarify himself.
https://www.youtube.com/user/BeanymanSports/videos