Well I've never seen Lex and Batman in the same room.

Batman questions anyone and anything and has a plan for any eventuality, PARTICULARY those that pose a global scale threat. It's nothing personal, it's just reasonable precaution. That's basically his true superpower. He also does trust Superman as a person, as a colleague and friend. I don't think he ever considers there to be a true risk that Superman turns on humanity of his own will. However, Superman is susceptible to mind control, to magic, to unpredictable forms of kryptonite. And he is not the only living Kryptonian in existence either. It would be stupid not to plan for such threats.

Lex depending on the version, may or may not think that Superman actually poses a willful threat to humanity. But even if he also trusts that Superman is what he appears to be, a selfless hero that only wants to help people, he probably hates that idea even more. He usually doesn't distrust Superman's intent. He hates what it says about and does to human-kind, and by extension, himself. He things depending on an alien demigod will make humanity weak and complacent. He thinks that Superman holds the Earth back from reaching their potential. That it permanently neuters them from become Supermen themselves. So he makes it his mission to ruin Superman however he can. If he can kill him, good. Not a problem anymore. If he can publically discredit him, sow distrust across the globe, that's good too, maybe better. People who distrust him won't depend on him and may, in fact, fear him. As a result they are more likely to better themselves, their technology, their science, to rival and fight back against Superman.

TL:DR: Batman takes precautions. Lex hates and attempts to kill or sabotage. They're not the same.

in the smallville series, he sees "clark" not trusting him(telling him hes an alien) so he develop some deep seated hate for him over that, like a grudge, which apparently carries on and may explain his adverserial nature in the future.

One of them questions superman, the other strives to destroy him at every opportunity.

All the DC billionaires think Superman is dodgy. Lex is a xenophobe who wants to destroy anything more powerful than him, Bruce sees a potential threat to humanity but respects the man. Oliver just thinks he's a pompous dick.

Unlike Bat or Super, people like Lex exist. Lex doesn't hate Super because he's powerful, he hates him because he's loved.

Lex's whole worldview is that the people who are smart and cunning should rule over those lesser. Super's existence contradicts this by serving the interests of people who are his inferior.

Lex must believe that Super is a fraud who doesn't deserve his position because otherwise Lex is the pretender.

The difference is that Batman accepts all of the good Superman does as well as the bad. Batman judges his capability but does not act without good reason.

Lex Luthor wants to bury everything about Superman's actions and only show the public how dangerous he is because he is alien and ungovernable. Lex will constantly put into action any and all plans against Superman.

It's the difference between open carry and attacking a different demographic's church.

Billionaires stick together

For Lex Luthor, I think that he doesn't like Superman because he's a threat to him, not that he's a threat to people. Which may seem similar, but the end goals are completely different.

Has anyone told lex Luthor that Superman’s mother is named Martha

It's actually worse than that.

Batman actually keeps green kryptonite in the Bat Cave as an insurance policy, should Superman ever go rogue. He has contingencies for everyone in the Justice League except Wonder Woman.

Entire point, wnd the key to understanding the BvS is that batman is broken in it. He's not the hero in the story.