In short "Rietveld analysis" is a subset of quantitative XRD. There are many ways to approach quantitative analysis that require the measurement of diffraction intensity in some form, such as reference intensity ratios or internal standard additions. The problem with these approaches is that one, or few peaks, are measured and they do not take into account the real problems associated with preferred orientation or peak overlaps. Rietveld refinement is a general technique that models a known crystalline structure and then by least squares refinement adjusts parameters until the model converges on the data. It can be used to find accurate cell parameters for pure phases. The application of the Rietveld method to quantitative analysis is allowing multiple structures to simultaneously refine, not only the crystallographic parameters, but importantly the scale factor for each phase in the assemblage; it is this that results in a measure of percentage for each phase, and importantly an estimate of the error on the measurement. The benefits here are that the whole powder pattern is used, peak overlap problems are overcome and there are models that can cope with preferred orientation. This method requires that you have good data, know all components in the system (since it normalises the results) and that you have good quality structural models for all phases. It is not a black box method and user input and interpretation is required since it is very easy to produce very high precision results that are meaningless!

In short "Rietveld analysis" is a subset of quantitative XRD. There are many ways to approach quantitative analysis that require the measurement of diffraction intensity in some form, such as reference intensity ratios or internal standard additions. The problem with these approaches is that one, or few peaks, are measured and they do not take into account the real problems associated with preferred orientation or peak overlaps. Rietveld refinement is a general technique that models a known crystalline structure and then by least squares refinement adjusts parameters until the model converges on the data. It can be used to find accurate cell parameters for pure phases. The application of the Rietveld method to quantitative analysis is allowing multiple structures to simultaneously refine, not only the crystallographic parameters, but importantly the scale factor for each phase in the assemblage; it is this that results in a measure of percentage for each phase, and importantly an estimate of the error on the measurement. The benefits here are that the whole powder pattern is used, peak overlap problems are overcome and there are models that can cope with preferred orientation. This method requires that you have good data, know all components in the system (since it normalises the results) and that you have good quality structural models for all phases. It is not a black box method and user input and interpretation is required since it is very easy to produce very high precision results that are meaningless!

The major question is: what do you mean with quantitative analysis, and what with Rietveld analysis? There are several thing "quantitative" in XRD, strain, crystal size, texture or preferred orientation, phase fraction.... I assume you are talking about the last one, but you should formulate your questions more precisely!

The other aspect is related to Rietveld analysis. I know that this is nowadays a synonym for full pattern refinement. This is nothing else than using a least square algorithm to compare experimental data with theoretical simulations by changing in a more or less clever way some parameter we assume to be relevant for the result. Originally, Rietveld proposed to use it for crystal structure refinement if no sufficient single crystals are available. As far as I know it took about 20 years until somebody had the idea to do the same with a mixture of phases (least-square): use the full diffractogram but does not change the structure at all. This remarkably reduces the number of free parameters, but because of peak overlapping, correct structure models etc it does not make a good analysis an easy going work, as Ian already pointed out.

Anyway, I am personally not very happy about this blurring of terms, especially because there is no real reason. From my point of view it looks easier to me to express with Rietveld EXCLUSIVELY crystal structure refinement. Then with full pattern refinement the "simple" scaling of phase-relates subdiffractograms is meant in order to find the best match with respect to the experimental signal.  We shouldn't also overview that least square as tool is much older and really not the specific feature of a Rietveld refinement. It is used everywhere... Rietvelds achievement was to realize that one does not have to give up if there are no single crystals available. Even a simple one-dimensional signal can be sufficient to refine a structure. But one needs to have a good model. Ab initio solutions are still a bit tricky but also possible nowadays.

I hope you finally understand that a correct formulation of a question is extremely important. It is sometimes more important than finding the answer.