Concepts of Calculus

1.) "Finding the limit of a function" is basically saying, "What is the output approaching to as x gets closer and closer to a number?" while "plugging in c for x" is saying "what is the output when x = c". The limit may be different from when you plug in c for x.
When these two concepts are the same is when you are finding the limit of a continuous function.
For example,
f(x) = 4 - x^2 at x = 2
As a continuous function, the domain has no restrictions thus x can be any number therefore
the limit as x -> 2 of f(x) = f(2)

2.) Finding the derivative is basically finding the slope of the tangent line thus for both, you use your change of y over your change of x.
The difference is that finding the derivative is finding the slope of a curve at a point while finding the slope of a line, you are only finding the slope of that specified line.