Consider a possibly nonlinear operator $F : \mathcal{P} \to \mathcal{Q}$, where $F(p) \in \mathcal{Q}$ represents, for example, the response of a physical system depending on a parameter $p \in \mathcal{P}$. Many-query problems -- such as stochastic simulation, optimization, or inverse problems -- become challenging when evaluating $F$ is computationally expensive, for example when it involves solving a partial differential equation (PDE). This motivates current research on constructing efficient operator surrogates that reduce computational cost while maintaining an acceptable approximation error. When $\mathcal{P}$ and $\mathcal{Q}$ are separable Hilbert spaces, inputs and outputs can be approximated in suitable finite-dimensional linear subspaces. Operator surrogates can then be constructed by approximating the mapping between the coefficients of the basis representations of input and output using either neural networks or polynomial methods. For both approaches, convergence results exist. However, for neural networks, these results often only indicate existence without practical guarantees, whereas polynomial surrogates yield constructive algorithms with deterministic error bounds. Key questions remain: does one method outperform the other, are there specific advantages or disadvantages to each approach, and do these depend on the application? This work addresses these questions through a comprehensive empirical comparison of polynomial and network-based surrogates across a range of PDE-driven applications. Additionally, we systematically examine how surrogate performance depends on the smoothness of the operator inputs. Our results reveal several patterns that are qualitatively similar across our test problems. In terms of accuracy relative to the number of training data, polynomial surrogates often outperform neural networks. Their relative accuracy advantage increases with the smoothness of the inputs. Network surrogates, in turn, excel in evaluation speed, another critical metric for operator surrogates. These findings not only help to select the most suitable surrogate for specific applications but also suggest to construct hybrid surrogates that balance speed and accuracy. Finally, this study is complemented by a comparison with more advanced and conceptually different neural network architectures, such as Fourier Neural Operators and Derivative-Informed Neural Operators.