Each of these adjectives is used to indicate horizontal extent. Broad is preferable when the word it modifies is a surface or expanse viewed as such (broad stream, broad field, broad shoulders). Wide is preferably used when the sense of space is stressed (the table is 4 feet wide) or when the distance across a surface is mentioned indefinitely (The lake is wide at that point). These words are often interchangeable, but idiomatic usage normally prevails: "wide mouth," "broad grin."
Dictionary of problem words and expressions. Harry Shaw. 1975.