The main difference between qualitative and quantitative research methods is the approach. With qualitative research methods, you're starting with a specific observation before moving to a more generalized one. In other words, you start with a specific fact and you seek to generalize it to understand it, so you can apply the information to a large group of people. This means that you have a specific topic like: one out of every 100 people suffers from a certain disease. Then you tear the observation apart looking for patterns before making a general hypothesis. However, quantitative research methods take a general approach with a hypothesis, then tests the theory to find data that either proves or disproves it. For instance, you theorize that heart disease is the leading cause of death in America, then work to prove or disprove that hypothesis.
Another huge difference between qualitative and quantitative research is compiling research data. For instance, after you have created your research project, you must decide the best method to collect data. If you choose to use surveys or polls, then you are employing the quantitative method. A survey consists of creating a number of questions for people to answer. However, a qualitative method uses a variety of ways to collect data. Therefore, you may interview experts, review documents, case studies, personal observations or conduct a focus group.
Math verses text is another distinction between the qualitative and quantitative research methods. Quantitative research is steeped in numbers. For instance, the surveys and polls require fixed number response options such as one through 10. Also, you'll use statistical tests for analysis. However, qualitative research method is based in text with very limited numbers---if any. This means that qualitative research seeks to explain through words, not statistics, the outcome of the hypothesis. For example, when writing your qualitative research report, you might explain the viewpoints of people in your focus group that experienced heart disease in some quotes, but mainly through an interpretation of data.