Currently, RadRichTextBox only imports from HTML documents tables with spacing between cells, but it does not export tables with such style. It seems like the border-collapse: collapse value defined in the Table Normal style overrides the border-spacing value, which is applied to the cells.
As a workaround, the document styles can be omitted from the exported HTML and the styling can be exported inline be changing the ExportSettings of HtmlFormatProvider:
provider.ExportSettings.StylesExportMode = StylesExportMode.Inline;
provider.ExportSettings.StyleRepositoryExportMode = StyleRepositoryExportMode.DontExportStyles;