We consider identifiability of partially linear additive structural equation models with Gaussian noise (PLSEMs) and estimation of distributionally equivalent models to a given PLSEM. Thereby, we also include robustness results for errors in the neighborhood of Gaussian distributions. Existing identifiability results in the framework of additive SEMs with Gaussian noise are limited to linear and nonlinear SEMs, which can be considered as special cases of PLSEMs with vanishing nonparametric or parametric part, respectively. We close the wide gap between these two special cases by providing a comprehensive theory of the identifiability of PLSEMs by means of (A) a graphical, (B) a transformational, (C) a functional and (D) a causal ordering characterization of PLSEMs that generate a given distribution P. In particular, the characterizations (C) and (D) answer the fundamental question to which extent nonlinear functions in additive SEMs with Gaussian noise restrict the set of potential causal models and hence influence the identifiability. On the basis of the transformational characterization (B) we provide a score-based estimation procedure that outputs the graphical representation (A) of the distribution equivalence class of a given PLSEM. We derive its (high-dimensional) consistency and demonstrate its performance on simulated datasets.