The RNA assay contains the raw counts, and if you use their older count normalization method (not SCTransform), the normalized and scaled counts. The integration assay is created after normalization and integration, as detailed in their integration vignette. The counts here are slightly adjusted so that cells that are (probably) similar between samples and technologies will cluster closer together with PCA and UMAP dimension reduction.

Hi,

You're setting your default assay as "integrated" or "RNA". A Seurat object has many data sets, usually the "RNA" data slot contains multiple data sets: https://github.com/satijalab/seurat/wiki/Assay

I do not remember quite well, but I believe that If you performed integration, apart from the "RNA", you'll have also "integrated" slot, with possible multiple data sets, like gene expressions tables normalized, transformed etc.

When you do - DefaultAssay(combined.data) <- "integrated" - you're setting the default assay "integrated". This means when you call other Seurat functions they will use this "integrated" slot data. If you want to use the "RNA" instead you do: DefaultAssay(combined.data) <- "RNA"

I hope this answers your question.

António