It sounds like you have dual citizenship? You don't really "use" a passport to leave a country, at least not in an immigration sense, although the airline or ship operators will want to see one as proof of ID, and if you prebooked and gave passport details, you would have to show the right one.
Legally, US citizens are required to use a US passport when entering the US and to have one in their possession when departing the US. (there is no formal immigration control when US citizens depart the US).
The UK requires you to use the UK one when entering the UK. Beyond these, you have a choice. Since France does not require either US or UK citizens to get a visa, I cannot see that it really matters which one you use on entering or leaving that country, except as mentioned above.