They're just "WGS84" as defined by different software packages. There's no difference in practice. Although you get the warning message in ArcMap, you can ignore it. Even setting a custom geographic transformation, won't change your data's coordinates.
The Project Tools (vector/raster) won't let you convert between them, because they are seen as different. You would have to use the Create Custom Geographic Transformation tool (9.2 and up) to define a transformation. Use geocentric translation method and leave the parameters set to zeroes.