For a complex function defined on a domain, "differentiable", "holomorphic" and "analytic" are equivalent, but the words emphasize different features of the function:
"Differentiable" emphasizes that the function can be differentiated (well, duh).
"Analytic" emphasizes that the function can be specified by power series in a neighborhood of every point.
"Holomorphic" emphasizes the phenomenon that the values of the function on some small subset can determine its values on the entire domain.
Differentiable and analytic have different generalizations to other function spaces (for example, among real functions the "analytic" ones are a strict subset of the "differentiable" ones). I think I have heard "holomorphic" only about complex functions.
"Conformal" is almost the same as differentiable/analytic/holomorphic, but includes the additional condition that the derivative of the function is nonzero everywhere in the domain. It emphasizes the geometric behavior of the function, and can be generalized to mappings between geometric objects that don't have the algebraic structure that the complex plane has.