quantitative refers to a number-based evaluation. Qualitative refers to a characteristic-based evaluation.
Quantitatively similar means that things are at the same general magnitude or size or concentration, or whatever numeric value you are using for judgement. Qualitatively similar means that, even though the size could be quite different, the same basic process or idea would apply (the thing would seem about the same without regard to size of the thing).
A black rock remains black no matter what it weighs. Qualitatively the same. A black rock and a white rock each weigh about 2 kg would mean that they are quantitatively about the same despite an apparent qualitative difference. However, if those two rocks had quite different densities, they would have quantitatively distinct volumes even if they had quantitatively similar masses (weights).
Naturally, similar and different are relative terms so you have to define what you mean by "similar" versus "dissimilar" when dealing with anything. How close to the same does a thing have be to be called "similar"?
That is, black can come in many shades, so it is possible for two black rocks to be judged qualitatively similar in a gross sense (both are "black") yet qualitatively distinct if I could distinguish between the two solely based on the color (distinct versions of black).
The general idea that distinguishes qualitative from quantitative is that qualitative is scale independent but qualitative is not. Sometimes, we use ratio numbers to "Normalize" the system and eliminate the scale factor. This would be why we use concentrations instead of mole numbers for many purposes. Number of moles depends on system size. Concentration does not. Both, however, are quantitative measurements (have a definite value).