I'd say that, if they don't cooperate, the results are usually less bloody.
They are probably also less likely to see imagine the end state of their cooperation to be, "After we are done with this job, I'm going to kill that SOB."
But a Chaotic Good roguish character might well have major problems cooperating with a standard LG Paladin.
And vica versa. The tough by the book cop and the vigilante don't always have to get along, or even see the other as a 'good guy'. In theory, the tough by the book cop might see the vigilante as being just as bad as the crime boss - failing to declare 'Chaotic Good' to be 'good' at all. Indeed, the crime boss and the by the book cop might have as much to cooperate in, as the by the book cop and the vigilante do. It's not necessarily clear that the by the book cop wouldn't cooperate with the crime boss against the vigilante, if he thought he could prove in court that the vigilante was a criminal but could not yet do so in the case of the crime boss.
An iconic example here is Jean Val Jean and Javert from Les Miserables. One reading of Javert is that he's simply Lawful Neutral with the law before every thing and nothing but the law. But it's also possible to read him as a Lawful Good with a deep character flaw. If you read the book, Javert comes off somewhat more nuanced than he does in the opera. His last act is to write a letter asking for a series of reforms in the judicial system and it's possible to read him as someone that seriously believes he's rooting out cruelty, injustice, and making not just a more efficient state - but a better world. But, despite having seemingly common aims, of course the two find it impossible to cooperate. And indeed, despite the fact that he's continually breaking his parole, it's possible to read Jean as a neutral good character - willing after his reform to submit to the law and even an unjust one, but only after and secondary to other priorities. He wants to turn himself in, but keeps finding he has other responsibilities that come first. And by that reading, it's possible that after his reform both are Lawful Good, but weight 'law' and 'good' and the priority of their duties - duty to God, duty to Country, duty to Family, duty to the Oppressed, etc. - differently. Again, it's not even clear that lawful good characters cooperate.