How does calculus different from algebra?
In general algebra is concerned with abstract ideas. Starting with variables itself, going through structures as groups or rings, vectors, vector spaces and ending on linear (and non-linear) mappings and many more. Also, algebra gives theory to many important tools such as matrices or complex numbers.
Calculus, on the other hand, is concerned with concept of tending meaning: being very close to something yet not being something. Out of this concept, mathematics created 'limits' and 'derivatives'. Also, Newton and Lebniz - fathers of calculus - thought of concept called 'anti-derivatives' which is integral.
On the other hand, calculus was concerned with areas under curves. Or rather areas in general. It is why since Aristotle people were trying to describe area under the curve by using rectangles. However, full mathematical formalism were created in 18th century by Riemann.
What was inspiration for Newton? Geometry. It was rather physics for Leibniz, as far as I can remember.