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RasterStack and RasterBrick objects are quite similar. However, RasterBricks have a shorter processing time than RasterStacks and RasterStacks are more flexible (e.g. with RasterStacks pixel-based calculations on separate raster layers can be performed).
In both multi-layered object classes, individual layers must have the same spatial extent and resolution. That is, individual layers must represent the same locations with the same level of detail.
In all 3 classes the data can be loaded in memory or on disk depending on the size of the grid(s). Raster objects are typically created from files, but even RasterBrick objects can exist entirely in memory.
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