The major point of comparison between these two types of methods is that both can be used to answer research questions. Both are useful in many fields, particularly in the social sciences and other disciplines that deal with human behaviors and attitudes.
From there, however, the two types of methods diverge. Quantitative methods are generally good for very closed-ended questions that can be reduced to numbers. For example, a political scientist might ask what percentage of white people vote Republican in presidential elections. A sociologist of health might ask what percentage of poor people get to see a doctor regularly. These are relatively narrow questions that can easily be put into numbers.
By contrast, qualitative methods are better for more open-ended questions that cannot be easily reduced to numbers. These methods are better for understanding why more white people vote Republican. They are better for understanding how poor people feel about the quality of the health care they get.
Both of these types of methods are important, but they are useful for very different kinds of questions.