Whats the difference?
As the term suggests. Quanti-tative research is the research which bases its conclusions on the quantity of respondents.
On the other hand, quali-tative is about the quality of the findings from a small, representative group of people
Quantitative research is the kind if research we are traditionally, conventionally exposed to. It revolves around the number of people you can get to answer your question(s). Further to which, number-crunching comes to play. A knowledge of statistics is crucial in deriving results.
We frequently come-across reports based on quantitative research.
"64% of urban Indian women feel they are discriminated against in office"
(That is just an
arbitrary piece of information I created. I don't know if there is in any truth in it.) Going by my
instincts, I don't believe there to be any truth in it. We can't rule out the
possibility of a small section feeling that way but I surely don't think that
more than half of the urban, working woman in India would think that. 
Quantitative research is like money. Its all about 'how much?'
Qualitative research is about being in touch with your instincts rather than your mathematical abilities
Qualitative research, like love is more about the qualityWhen do we use quantitative research?
When you want specific numbers.
Saying, "most urban, working women feel discriminated against" can never sound as satisfactory as "64% urban, working women feel discriminated against" Of course, the source of the information is of importance. You can depend on any top research agency with the requisite facilities to do quanti research.
Quanti findings is more for brand marketers to understand market drivers etc.
When do we use qualitative research?
When you want insights and a deeper understanding of the consumer. Perceptiveness and the ability to follow instincts is of primary importance in the people that constitute a quali research team.
e.g if someone frowns at a brand logo, the quali moderator should be able to tell whether they are frowning at the logo itself or at the bad after-sales-service he/she gets from the the brand it represents.
To change the logo may not be the right recommendation then would it?
Quali findings are for advertisers who need consumer insights and consumer understanding.
PS: The above scribbles are simply based on my experience in advertising. A professional researcher would give a better insight into various kinds of research and the applications of it.