Both Movies Begin In A Grizzled, R-Rated Future Earth
Logan is set in the year 2029. Children Of Men is set in the year 2027.
In both stories, our main character works a menial job in a barren future devoid of resources. Animals are going extinct (in Logan they mention there are no longer tigers) and while civilization still exists, it does so in an aura of dusty desperation.
Violence is abundant, which is why both films require an R-rating. The difference is that Children Of Men knows how to use their rating, whereas Logan spends the first 20 minutes shoehorning more "fucks" than a little kid who just discovered the word. At one point a lady flashes Logan her boobs and it adds nothing to the plot beyond the studio saying "Hey look, we're rated R now!"
Both opening scenes feature an act of limb-tearing violence. The difference being that Logan's is a sequence where he brutally murders hubcap thieves (seems a bit much, but sure) versus a bomb going off in Children Of Men. One is horrifying while the other is played for action gasps.
Spoiler: This woman does not show her boobs.
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.