Qualitative methods are things like interviews and case studies, in which you have a small number of people and get a great depth of information from each one, learning their whole story and all its ramifications.
Quantitative methods basically always mean statistical analysis in some form, either using statistics that are already collected (such as by the government) or by collecting new statistics through surveys or in an experiment.
Surveys actually somewhat straddle the line between qualitative and quantitative methods, depending on whether the survey is a very long, detailed survey given to a small number of participants or a very short, simple survey given to a large number of participants.
A couple of advantages of qualitative methods to consider:
1. You're not very dependent on a particular hypothesis. You can explore the subject in great detail and try to formulate new ideas.
2. Detailed, holistic information allows you to place findings in context and know that you aren't missing out on important information or pigeonholing your participants into your own preconceived ideas.
A couple of advantages of quantitative methods to consider:
1. You can get much more precise, replicable results. Other researchers can check your math, repeat your study, or even conduct their own studies to see if they get the same results.
2. Quantitative results are often much more useful in the real world than qualitative results. Often we already know that there is some effect of a particular factor, but what we really want to know is which factors have the biggest effect, or what the effects of a given policy change might be. This is particularly true in economics, where the difference between a good tax policy and a bad tax policy can be as simple as a few percentage points.
Generally speaking, qualitative methods are best when we first approach a new subject and really don't have our bearings. They help us formulate hypotheses. Then, once we have established sufficient background knowledge about the subject, quantitative methods allow us to precisely test our hypotheses and determine which ones are correct.
Of course, each advantage for a given method can be articulated as a disadvantage for the other method; quantitative results being more precise means that qualitative results are less precise, etc.