I don’t know if there are any differences left (save the DH). AL umpires used to wear enormous chest protectors that more or less forced them to be square behind the catcher. The NL umpires wore thinner chest protectors under their jackets and tended to crouch on one side or the other of the catcher.
The weak-hitting pitcher affects the strategy of the NL game. You have a pitcher who has been effective come up at a crucial part of the game. Do you keep him in or pull him for a pinch hitter? I’ll confess, I’m a life-long NL fan. The AL with its DH and the change it brings to the basic strategy of the game just doesn’t appeal to me.
Perhaps it’s a case of just getting older. The way things used to be done are often seen as better. I’m old enough to remember college football games where players played both offense and defense and quarter backs pretty much called their own game. It’s why I find I enjoy watching rugby on TV more than football.
There's only other area where the AL and NL had different rules as recently as the last few years: the AL had a curfew for night games, while the NL did not. In an AL game, they were not allowed to start an inning after 1:00 AM unless there were special scheduling circumstances involved (typically if it was the last game of the season between the two teams, or one or both of the teams had their remaining off days committed to rescheduled games for rainouts). The NL didn't have a curfew rule, which sometimes resulted in long games ending early the next morning.
I remember an infamous game the Mets and Braves played in the 1980s one Fourth of July. There was a fireworks display scheduled after the game, which ended at almost 4:00 AM after 19 innings and two rain delays. They set off the fireworks anyway, causing all kinds of calls to the police department from people who thought there was some kind of industrial explosion early in the morning. LOL.