There are many advantages to classification, both in science and "out" of it.
Classification allows us to see relationships between things that may not be obvious when looking at them as a whole. For example, by classifying participants in a study according to many different elements of their lives (height, age, weight, etc) we might be able to identify a relationship between some of those traits and the outcome of their lives. This is a pretty common practice in education as well; for example, there are dozens of variables that may affect how a student performs in an academic setting, and classifying those variables allows us to evaluate which ones have the greatest impact, and which can be controlled.
Categorizing things also makes it easier for us to make subjective judgments about the worth of different things. One such categorization is triage, which is a way of evaluating who should receive medical care first in the event of an emergency. Triage traditionally consists of categorizing patients according to three tiers of injury; serious, moderate and light, with serious patients receiving care first.
A disadvantage to classification is that many of the classifications themselves are based on subjective judgments, which may or may not be shared by everyone participating. This would lead to differences in perceived value. For example, if someone classifies attractiveness according to height and hair color, this may lead not only to negative social impacts, but because it is based on relatively superficial traits, it may also lead to reinforcement of poor value systems and diminished fitness.
Another problem is that humans are the ones doing all of this classifying and categorizing, for our own benefit; the natural world doesn't function according to categories. Thus our own knowledge can sometimes create an artificial limit on our ability to correctly categorize something. This is evidenced by the frequent disputes and reorganization of evolutionary relationships, or by the false classifications caused by fraudulent or mistaken research, such as the false link between autism and vaccines.