Significance of requiring a nowhere vanishing derivative The following is from J.S. Milne's notes on modular forms.  He gives a definition of compatible coordinate neighborhoods.

What is the significance of requiring that the derivative be nowhere vanishing?  

No significance. 
If you assume only that they are holomorphic, interchanging the role of $i, j$ gives 
$$z_j \circ z_i^{-1} :z_i (U_i\cap U_j) \to z_j(U_i\cap U_j)$$
is holomorphic. But this is the inverse of $z_i \circ z_j^{-1}$. Thus both of them has non-vanishing derivatives, which can be checked using chain rule. 
