Quarrels tend to be short and sharp: wrangles take longer, because a wrangle will involve argument and probably even a bit of logic.
Sometimes, indeed, a wrangle is almost friendly: if you're wrangling you may use charm and possibly a little encouragement to get what you want. On the other hand you may just argue people into submission.
If you're in the American West or in Canada then a wrangler will be trying to persuade a whole herd of individuals to do as he says. Still, they'll all be horses or cattle, so argument won't have much to do with it.
A Senior Wrangler is someone who's been the top undergraduate in Mathemathics at the University of Cambridge. Jacob Bronowski and Lord Rayleigh are among them.
An ordinary (but obviously not that ordinary) wrangler can be someone who's got a First in Maths from Cambridge, or it can be an animal herder (especially someone who looks after the animals on a film set).
Now it's turned warmer here in England I'm going to have to start wrangling slugs: rounding them up and taking them a long way away, that is, not grooming them for show business.
And I can assure you that I won't be listening to a single word of argument from any of them.
Thing To Do Today: wrangle. This word comes from the Low German wrangeln, which is a frequentative of wrangen, to struggle.