The main difference between the two is the anechoic chamber is built without any echoes and is not room influenced at all. Quasi-anechoic measurements are not accurate below the midrange where the room starts to dominate with this technique. So the amount of accurate information you can gather is more limited with the quasi-anechoic measurement.
Neither technique can measure low frequencies accurately. This is why subwoofers are measured outside.
Different db levels show at what SPL the frequency sweep occurs at. If there are deviations in the frequency response at increasing sweep levels it is a sign the speaker is stressing aka compression.
