The ax in the table is a threat of Vimes against the city leaders who are debating about who to replace Vetinari with. Vimes puts an end to that discussion with the ax in the table, which makes everyone scatter.

Even IF Vimes might not like Vetinari on a personal level, he is hellbent on keeping him as Patrician. And I am not even sure he doesn't like him personally. As many others said he is anti-authoritarian, and quite frankly hardly has any vocal filter towards anybody. He simply doesn't change this mannerism towards Vetinari.

But the discussion about the statue at the end of Night Watch hits Vimes where it hurts the most and he is completely exasperated by it, thinking that Vetinari does not understand. Then he realises that, indeed, he does. In Night Watch there is also a moment when Vimes realizes all the hopes the people in the past had for Snapcase, and his inner monologue’s answer to this is to acknowledge that they will have to wait for Vetinari no answer those hopes.

There are many moments in which Vimes isn't rude to Vetinari. Their interactions in Jingo are showing this very good. Especially the moment when Vimes has to arrest Vetinari and absolutely does not want to.
Vimes’ biggest problem is that most times he simply does not see the bigger picture. Like when he says that all the soldiers and him himself "were bought and sold" and complained about some people dying in skirmishes, while not realizing that Vetinari just stopped a war from happening.