What is the difference between Qualitative marketing research and Quantitative marketing research?

Qualitative research studies 'notions' rather than creating statistics.  Qual will focus your hypotheses and thin-out your you ideas.  It's step 1 in the research process.  Typically, you will ask many open ended questions to gain insight and help create hypotheses (or questions) to test later.

Quant research (step 2) is the testing of the key questions created during qual.  It is the statistically valid survey of respondants that give answers to the key questions.