which is better for an american family to live in temporarily? which is more convenient for exploring europe? how far is each to other major cities, etc?

Brussels is much better located from a transit hub sense - Paris a few hours by train as is Germany. Amsterdam is a dead-end for transit. Whether which city is best for a family would entail all kinds of discussion, from Amsterdam's legalization of soft drugs affecting teen-agers, family values, etc. Physically Amsterdam is one of Europe's most beautiful cities, Brussels one of its least. Yet nearby Belgian towns like Leuven are gems.

Brussels actually has some quite attractive neighborhoods. Unfortunately, most tourists don't venture far enough from the central core to find them.

Amsterdam is FAR more interesting and most of its residents seem to speak English very well, if that's at all an issue.
And I disagree that it's a transit dead end. Train connections to Germany and Scandinavia a better from A'dam, and the Thalys gets you to Brussels or Paris quickly and efficiently.
And as an air hub, A'dam blows Brussels away, both for international and intra-European connections.

A lot of the decision will probably revolve around housing. If housing is provides and/or paid for by the employer, then Amsterdam would probably be more fun in the long run. But if they have to find their own housing or pay for it themselves, then Brussels holds the trump card. Brussels is used to vast numbers of people staying for a few months (EU assignments, etc.) so has an excellent stock of short term, furnished housing at good rates. Housing in central A'dam is hard to find (esp. if you want more than one bedroom) and pretty expensive. Food is better in Brussels, too, but that's hardly a major consideration.
While A'dam's airport is superior, I wouldn't say that train connections are better. I had no problem going all over Germany from Brussels. High speed ICE and Thalys trains serve Brussels frequently.
Won't the wage earner need some kind of work permit? Belgium is pretty quick with those who have bona fide employers offering to sponsor them. My work permit came through in less than three weeks. Don't know how long that would take in the Netherlands.