Algebra VS Calculus

1. What is the DIFFERENCE between finding the limit of a function at x=c and actually plugging in the number x=c? When are the two cases the SAME?
When you are try to find the limit at x=c you are also find the output and the value of f(x) when it gets closer to c. There could be holes at x=c, but for the limit to exist the hole has to be removable. This is calling a Removable Discontinuity.
The two cases are the same when the graph of the function is continuous.

2.What is the SIMILARITIES between finding the derivative and finding the slope of a line. What are the DIFFERENCES between the two?
The similarities between both of them is that you have to use the change y / the change x to find the slope.
The difference is that on the derivative you need to find the tangent line to find the secant line.