Amount of, number ofand quantity of have more or less the same meaning, but they are not interchangeable. For instance, you can say ‘There is a huge amount of garbage in the backyard.’ Or, ‘A number of issues need to be addressed during the meeting.’ You could also say ‘Only a small quantity of sugar needs to be added to the mixture.’ Or ‘Very large quantities of food are needed to solve this crisis.’
You use amount of for uncountable nouns. So, you would say ‘The amount of cement that you used is not nearly enough.’ because the noun cement cannot be counted. You cannot say ‘one cement, two cements’. The same goes for the sentence ‘He had an incredible amount of luck.’ You cannot say ‘one luck, two lucks’.
You use number of to talk about nouns that you can count. This means that you would say ‘The number of robberies has gone down in this area.’ because you can say ‘One robbery, two robberies’. The same applies to the sentence ‘This traffic sign should reduce the number of accidents.’ The noun accident can be counted.
If you want to be more formal than amount of or number of, you can also use quantity of or quantities of. You can use them for both uncountable and countable nouns. This means that you could say the quantity of in combination with the uncountable noun salt in the sentence. ‘The quantity of salt in this product should be lowered.’ You could also say, however, ‘They found a considerable quantity of diamonds in that mine.’, when using the countable noun diamond.
Quantities of works in the same way. For instance you could say ‘We import large quantities of coffee every month.’ when using the uncountable noun coffee, and you could say ‘Large quantities of books will be needed to meet demand.’
And that is the difference between amount of, number of, and quantity of.