I would use 'value' in equations and formula such as
What value of x makes this equation true ... 2x + 3 = 17 ?
If F = 9C/5 + 32 ... what is the value of F if the value of C = 40 ?
I would use 'quantity' when measuring something such as
What quantity of sand is needed to make a 3:1 concrete mix if I have 25 kilos of cement.
I think you can always change 'what quantity of' to 'how much'; thus
How much sand is needed .......................?
EDIT: I've just looked on dictionary.com and found this:
value 6. maths
a. a particular magnitude, number, or amount: the value of the variable was 7
b. the particular quantity that is the result of applying a function or operation for some given argument: the value of the function for x=3 was 9
Interesting that they use the word quantity whilst defining value.
4. Mathematics .
the property of magnitude involving comparability with other magnitudes.
something having magnitude, or size, extent, amount, or the like.
magnitude, size, volume, area, or length.
So perhaps the main difference is that value is used when you want to be specific, such as
I have a number of piles of objects. I know the quantity of objects in each pile. Which value will satisfy this specific requirement ..............