What an interesting question. I have often thought about the real meaning of "citizen", but rarely about the word "subject" because it feels so old-fashioned. There are so few "subjects" remaining in this world...
Or are there? I see the literal meaning of the word "citizen" as the inhabitant of a city. The medieval city-states of Europe were integral to the development of trade and commerce, as I vaguely remember it from my study of history at university... so I always think of citizenship as a kind of evolution away from feudalism and blind obedience to royalty.
Citizenship is usually seen as a two-way street, where you have responsibilities to government and the community, as well as receiving protection and civil rights. So I would certainly prefer to be called a citizen than a subject. But I think that the word "citizen" is also starting to fall out of favour in some respects, as the term is increasingly associated with invasions of privacy and personal freedom, iniquitous taxation and government corruption. As a result, the meaning of the word "citizen" is coming closer to the meaning of "subject", because citizens are subject to so many rules and regulations - for example, rules governing freedom of speech are tightening so much that people are being jailed for hate speech when they expressly intended no offence. In this way I think there is a danger that citizens are becoming subjects.
Phew! A thought-provoking question on a topic that I think about a lot these days!