In this context sensitive means discriminating, perceptive of nuances, aware of subtleties, or having critical discernment, a depth of perceptiveness, or refined artistic judgment. It is used in this sense mostly when discussing judgments about, analyses of, and reactions to artistic productions such as paintings, music, and literature. Paying close attention is just a part of the meaning, as described above.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.