Holomorphic functions on an open set but not a domain

Yes, there are big differences. For example if $U$ and $V$ are two disjoint open sets then $f(z)=1$ for $z \in U$ and $f(z)=0$ for $z \in V$ gives a holomorphic function on $\Omega =U \cup V$. Its zeros have a limit point but the function is not identically $0$. This cannot happen when $\Omega$ is a domain. Any holomorphic function whose zeros have a limti point is identically $0$ in this case.

Every open set can be partitioned into at most countably many domains (the connected components of the open set). A function is holomorphic on an open set if and only if it is holomorphic on every connected component thereof (since being holomorphic is a local property). This basically tells you everything about the relationship: a holomorphic function on an open set is just a completely independent collection of holomorphic functions on domains. Otherwise said: if you understand everything about holomorphic functions domains, you immediately also know everything about open sets - and books using open sets as the default will explicitly say when they want connectivity.

You would need connectivity for the identity theorem. You wouldn't need it for any explicitly local theorems such as if you derived the Cauchy-Riemann equations from complex differentiability or noted the invariance of line integrals under homotopy of the curve. Sometimes you'll even want things to be simply connected, meaning that every curve is homotopic to the identity - for instance, for getting an antiderivative to a holomorphic function. There's a few flavors of these connectivity requirements - but it's not so meaningful which is used as the default.