When Michael asks Roth who gave the order to Kill Pentangeli, Roth gives his famous "someone put a bullet in his eye" about Mor Green. Obviously he knows Michael gave "the order," but then Roth who'd been praising Green for having "invented" Las Vegas says he didn't do anything about it because "this is the business we have chosen. Clearly, however he took this personally and held a personal vendetta against Michael for it,
DESPITE the fact that Green's assassination was strictly business. In GFI when Michael tells Green he wants to buy him out, he points out that Roth's casino was losing money. Green defensively asks whether Michael thinks he is skimming off the top (assuming Green was to Bugsy Siegel what Roth was to Meyer Lansky he was skimming). Michael says no, "you're just unlucky." Green then gets really angry saying how he took Freddie in when the Corleones were having a bad time and now that they are being chased out of New York they try to move him out. Michael corrects him and reminds him that he took Fredo in because "the Corleone family bankrolled your casino, and the Molinari family on the coast guaranteed hs safety.
If the Corleones actually bankrolled Green and he was losing money for whatever reason it would be bad business to let him continue. Moreover, Green goes on to imply that because he "tawlked" to Barzini, he could make a deal with him and still keep his hotel. In other words it would either be the Corleones or Barzini who would eventually control Green, and whatever Barzini promised, I guarantee if Green continued to lose money, Barzini would have moved him out as well. Bottom line therefore is that this was an extension of the war between the Corleones and Barzini and the other families, and had nothing to do with Hyman Roth. In fact according to Johnny Ola the one reason Roth survived as long as he did was because he always made money for his partners. If you follow that line of reasoning even Roth would not have backed Green if he was losing money for his partners.

Even though Roth admits the killing of Green was "bsiness," it is very clear his bad feelings for Michae were not about business, but strictly personal.