Both Characters Are Ex-Heroes Who Turned To Booze
Logan is fa-king grizzled. Nasty cat drains flammables faster than Truckasaurus, spending the first act wafting from bottle to bottle like a drifter genie. The film really wants us to know what kind of an alcoholic Logan is... which is strange when that information ultimately amounts to nothing.
Meanwhile, Clive Owen's character in Children Of Men is also an ex-hero (in this case an activist) who has sunk into the depths of whiskey sadness. After spending the film gulping from a half pint, his alcoholism comes to a subtle conclusion when he's forced to deliver a baby in less-than-ideal conditions.
"Are you sure your it's your water that broke and not my Johnny Walker bottle?"
In a moment of panic, C-Owes yanks out his familiar bottle, goes in for a sip, and opts to use it as a hand sanitizer instead. He doesn't make a big show of it, but it's enough of a gesture to see that his character is beginning to crawl out of his stanky rye pit. No such moment of catharsis exists with Logan, who simply stops drinking halfway through the movie like it never happened.
Both Characters Are Reluctantly Tasked With Protecting A Young Lady (By Another Lady Who Dies)
The plot of Logan kicks into gear when he is dragged into helping a woman harbor a young, quiet girl. The caretaker is then brutally murdered and so Booze-Wolf is forced to protect her from evil men who want to use her for their own sinister plans.
This is the exact same catalyst in Children Of Men, when Clive Owen is dragged into helping Julianne Moore harbor a young, also-quiet girl. Moore is then brutally murdered, forcing The Cliver to reluctantly protect her from bad guys who want to use her for their own sinister plans.
Only when Julianne Moore dies, it's unexpected and emotionally devastating for the main character. We spend time watching him buckled over in tears. He's helpless. In over his head. It's sad, awkward, and frightening to witness, like a clown suicide or Hannibal Lecter doing his taxes.
Just thank me for not making this a GIF.
But in Logan, the murdered woman is a total stranger. There's no moment of weakness or fear. Her death isn't emotional, because ultimately she's just there so Wolverine can find a perfectly edited video on her cellphone explaining the plot. Thank goodness she found time to sit down with iMovie and churn out a short-form documentary while running for her life!