Answer:
It depends where you are going to. Sometimes, places where both of these passports have visa-free access to, would treat you differently at the point of entry depending on your nationality.
Places where it's better to show a British passport:
the EU countries, Brazil (Americans need a visa for this one), Australia, New Zealand, U.A.E., and Turkey.
Places where it's better to show an American passport:
Canada (ironic! but they take it easier on Americans)