In several ways, they really aren't any different. A confirmatory bifactor model using IRT is really just a full-information method to deal with categorical data directly, rather than the limited information methods found in structural equation modeling (SEM). Had the observations been continuous, a structural equation modeling approach would be the correct way to analyze the model anyway. Bifactor models in the IRT literature tend to be very popular since they are widely applicable in ability testing situations, and have had special treatment of how the models are estimated, but fundamentally are the same as traditional factor analysis methods (IRT is best thought of a non-linear factor analysis of categorical data, but there are other benefits to it as well).
The limited information SEM approach when exploring these kinds of models for categorical data is still beneficial though since they aren't so computer intensive (they only deal with estimating the first and second moments, rather than all moments as IRT does). However, they require special kinds of estimation and correlation matrices, such as a tetrachoric or polychoric matrix, along with special kinds of estimators such as the WLSMV algorithm.