Brawny is quite old-fashioned here. It's not a word I would ever use, so I'm not that sure of the difference. But to me - muscular is more that the muscles are well-defined, i.e. that the person is not carrying much fat, as well as having muscles. In other words, brawny could mean strong, but not thin.

Brawny: Charles Bronson in anything, maybe Arnold Schwarzenegger. Better yet, think lumberjack. Not brawny but ripped: Christian Bale in American Psycho, etc.

I disagree with David, muscular and brawny are not synonyms. A guy can be 1.85m/6'1" and weigh 60kg/130lbs and be muscular, but nobody (at least nobody I know) would describe a guy with that build as brawny.

They mean the exact same thing, but muscular would be more common. Though in replacement of these words, you would say strong.