What is the difference between sigma and integral?

Sigma is a discrete sum, a sum with steps. Eg. add the numbers from 1 to 10 or add the numbers 1/2, 1/4,...

A sigma always has a concept of a next thing to add, even if the list of things goes on forever.

An integral is a continuous summation. It is a summation in that we are adding up the area under the curve, for example, but it is continuous in that because we are adding things of arbitrary smallness it's not really possible to always point to the individual terms that are being added because they become some kind of continuous blur. Instead we use some mathematical technique (integration). But still it is a kind of summation.