complex differentiability
Complex differentiability at a point z_0 is the property that the difference quotient actually has a limit — that f has a well-defined complex derivative f'(z_0). Said another way, near z_0 the function behaves like multiplication by a single complex number: f(z) is approximately f(z_0) + f'(z_0)(z - z_0), and that linear part is multiplication, not just any linear map.
Compare with the real plane. A map of (x, y) is real-differentiable if it has a good linear approximation, given by a 2-by-2 Jacobian matrix with four free entries. Complex differentiability is MUCH narrower: the only 2-by-2 matrices that act as multiplication by a complex number a + i b are those of the special rotation-scaling form [a, -b; b, a]. Demanding the approximating matrix have exactly this form is the same as demanding the Cauchy-Riemann equations u_x = v_y and u_y = -v_x.
Because of this, complex differentiability at a single isolated point is almost worthless; the magic only appears when f is complex-differentiable at every point of an OPEN set (then we call it holomorphic). On an open set this one condition cascades: the function is automatically infinitely differentiable, conformal where f' is nonzero, and locally equal to a convergent power series. None of these free upgrades happen for a merely real-differentiable function.
Writing f = u + i v, the map (x, y) -> (u, v) has Jacobian [u_x, u_y; v_x, v_y]. Complex differentiability requires this matrix to be of the form [a, -b; b, a], i.e. u_x = v_y and v_x = -u_y — and then f'(z) = a + i b = u_x + i v_x.
Complex differentiability = the Jacobian is a rotation-scaling, which is precisely the Cauchy-Riemann condition.
Differentiable at one point only is nearly useless; the deep theorems all need differentiability on an open set. That is the difference between 'complex-differentiable at z_0' and 'holomorphic'.