The most important difference between these two is that quantitative research tries to build mathematical models using numerical data while qualitative research is based on broader questions and on data in the form of words. In some social sciences there is a great deal of conflict as to which of these methods should dominate.
For example, many political scientists today try to do quantitative research. They focus on trying to model things like Congressional behavior. This stands in contrast to more traditional political scientists who try to do things like interviewing members of Congress and other decision makers to try to determine how things work in Congress.
Quantitative research has the advantage of being (or at least seeming) more objective and scientific. Qualitative research has the advantage of looking in more depth and nuance at human behaviors rather than trying to reduce them to numbers.