Sensitivity analysis is used to assess the sensitivity of model output (or other components) to changes in input, parameters and other relevant components of the model. In this way, you can identify for which inputs, parameters etc. the model output is sensitive and this information can be used in e.g. calibration of models or uncertainty analysis. In calibration, you generally want to include the most important parameters (for which the model output is most sensitive). In uncertainty analysis, you want to focus on those inputs, parameters etc. which are inluencing the model output most (again, for which the model output is most senstive).
Extreme conditions tests can be used as a validation method for models. By exposing models to extreme conditions, one can verify whether the model provides reasonable outputs. For instance in hydrological modelling, extreme rainfall conditions can be used to verify that the model is generating a lot of (surface) runoff and responds rather linearly to extreme conditions (i.e. at some point all rainfall will contribute to the runoff since the system will become completely saturated). As such, an extreme conditions test is a qualitative validation method and might be regarded as a very specific type of sensitivity analysis.