Thanks for your comments, Morgan and Andrew.
So now I understand the difference comes from the fact that RT falls back to sRGB, if no profile is found. This explains the difference. Yes, I have seen that sRGB profiles are not all the same.
But one thing is still strange. As I had written earlier today, opening the image developed in RT again in RT gives the same colours as during development, which is what one expects. Opening this image developed in RT with RT_sRG embedded in PSE does give different colours. Only if I turn off colour management in PSE (i.e. it ignores the embedded profile) I do get the same colours as in RT. Does this mean, that RT somehow knows that the image was already developed and is in RT_sRGB (meaning nothing is to be done), whereas PSE does not realize this and applies the embedded profile again, which ruins the colours?
Best wishes
Hermann-Josef