Structural reliability methods are nowadays a cornerstone for the design of robustly performing structures, thanks to advancements in modeling and simulation tools. Monte Carlo-based simulation tools have been shown to provide the necessary accuracy and flexibility. While standard Monte Carlo estimation of the probability of failure is not hindered in its applicability by approximations or limiting assumptions, it becomes computationally unfeasible when small failure probability needs to be estimated, especially when the underlying numerical model evaluation is time consuming. In this case, variance reduction techniques are commonly employed, allowing for the estimation of small failure probabilities with a reduced number of samples and model calls. As a competing approach to variance reduction techniques, surrogate models can be used to substitute the computationally expensive model and performance function with an easy to evaluate numerical function calibrated through a supervised learning procedure. Both these tools provide accurate results for structural application. However, particular care should be taken into account when the reliability problems deal with high-dimensional or strongly nonlinear structural performances since the accuracy of the estimate is largely dependent on choices made during the surrogate modeling process. In this work, we compare the performance of the most recent state-of-the-art advance Monte Carlo techniques and surrogate models when applied to strongly nonlinear performance functions. This will provide the analysts with an insight to the issues that could arise in these challenging problems and help to decide with confidence on which tool to select in order to achieve accurate estimation of the failure probabilities within feasible times with their available computational capabilities.