I think that both movements need to be defined prior to determining which age we live in today.
The Neoclassical period was one in which the artists and authors of the time "looked to the classical texts for their creative inspiration in an effort to imitate classical form. The writers in particular drew on what were considered to be classical virtues—simplicity, order, restraint, logic, economy, accuracy, and decorum—to produce prose, poetry, and drama. Literature was of value in accordance with its ability to not only delight, but also instruct."
Romantics, on the other hand, emphasized "the importance of the individual’s experience in the world and his or her interpretation of that experience, rather than interpretations handed down by the church or tradition."
Therefore, each movement was distinctly different based upon their ideologies regarding tradition. The Romantics wanted to move away from being told what to do, while the Neoclassicists wished to bring back what had been done in the past.
Based upon this, I would say that our world today can be considered one of the following: purely Neoclassical, purely Romantic, a mixture of both Neoclassical and Romantic, or neither Neoclassical nor Romantic. While this answer may seem obvious (all, nothing, or both), the answer lies in the fact that not all people regard the world in the same way.
For some, they would love to return to a life where they are instructed in the classical ways of life. Others would simply like to live their own lives based upon their individuality and unlimited imaginations. That being said, there are some aspects of life where both, Neoclassicism and Romantic, ideals can live coherently; while in other aspects of life, neither Neoclassical nor Romantic ideals would be relevant.
Therefore, in the end, the answer to your question will simply never be able to be answered in a universal way. Not all people will agree on the type of age which we live in.