Citizen: 1. A legally recognized subject ornational of a state or commonwealth, either native or naturalized. 2. An inhabitant of a particular town or city.
Denizen: 1. An inhabitant or occupant of a particular place.
I would say the answer depends on how technical we're being.
A citizen of the United States is a legal resident who has been processed by the government as being a member of the United States.
A denizen of the United States is simply someone that lives there.
Technically speaking, one could never be, for example, a citizen of the Earth -- but we're all denizens of the Earth.
I can think of three types of use where denizen works, and citizen doesn't.
"Citizen" means a person who is not just present in a city or other conurbation, but at least potential part of its social body.
"Denizen" is a much more general word which is not limited to humans, and not limited to civilised or organised places. It often has a connotation of wildness.
"Denizen" is preferable to a sentient nonpartisan observer in self-description. The legal term "citizen" makes the claimant liable for the actions of corporate Person/actors.
I STRONGLY disagree with this current "top answer":
'A citizen of the United States is a legal resident who has been processed by the government as being a member of the United States.'
This is revisionist history and a disgrace to the very concept of citizenship.
A denizen is simply one who resides in a location (we agree here).
A subject is one who has been declared a member of a nation by a Ruler or State.
A citizen is one who has declared himself a member of a nation.
The moment we think otherwise we devolve into subjugation.
Citizenship arises from the individual alone.
A state cannot grant or deny citizenship. Only an individual can do this.
This change from top down membership in a community to bottom up membership in a community is the true meaning of:
e pluribus unum
novus ordo seclorum
1776 the very concept of being "subject" to a king through subordinate allegiance or conquest was OVERTURNED, In The Spirit of The Revolution, by the republican concept of consenting to being a "citizen".
I cite Ramsay Dissertation of 1789