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The connection is Normandy. The Bruces are actually a Norman-French family who married into the Scots royal family. However it's a location-surname from a place in France known contemporarily as "the brush(wood)", so there's not necessarily any family connection with Norman-Scots Bruces and the French Brousses. In French "brousse" now refers to wilderness or rugged rural areas, but it's also very similar to the Provencal word (bróusse) for a type of cheese with a similar name in Corsica and Italy, which may mean your Brousses have another origin entirely and merely had their name anglicized when they came to the US.

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