That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred to as regular functions.

Does analytic imply holomorphic?

A function with a convergent complex power series ∑ an(z − z0)n is called an analytic function. Analytic implies Holomorphic in the disc of convergence.

Can a function be analytic but not differentiable?

By definition a complex function is analytic at a point if it is differentiable not only at the point itself, but in its neighborhood. If you want formally, a function is analytic in z0 if there exists ϵ>0 such that the function is differentiable at every point z which satisfies |z−z0|<ϵ.