is there a difference between integrating a function between two limits and summing a function and if so where does the difference come from and when would you use each method in real life situations
There are multiple differences, but the main thing to note is that summation involves discrete values whereas integration involves continuous values.
Something else to note is that integration is just summation over infinite values.
As for in "real life" situations, if your professor told you to count the number of students in your lecture hall, you would count and report to your professor. However, if your professor told you to count how many water particles are in the ocean, then realistically, you would estimate the number of water particles in a smaller area and integrate all of those smaller areas. This will give you a close approximation of the total number of water particles.