Definition of analytic function – A function is analytic at c means that the Taylor series of the function converges to itself at point c.
In other words, at a point c, there exists a series that exactly equals to the function. For example, e^x is analytic for all x. Because the higher estimated value of e^x, Taylor series, has no difference with e^x.
Infinitely differentiable functions can be differentiate uncountable times and never ended. However, an infinitely differentiate function may not be analytic for all x. For example the function f(x)=e^(-1/x^2) for all x except for 0, while equal to 0 at x=0. This function is infinitely differentiable but not analytic for x=0.