The difference is important because some collations should distinguish between the two cases and put them in different places when sorting. Another difference is that in traditional fine typography sometimes different sorts were used for the two cases. The umlaut should have the dots closer to the letter body than the diaeresis. Yannis Haralambous writes about these topics in his O'Reilly book Fonts & Encodings, especially about the fact that Unicode can't really express this nuances and if I remember correctly that some software uses the order of combining diacritical marks to encode the difference.
If there are previous examples of the two rendered differently, I think that makes a pretty good case for extending Unicode.
...and kudos to salon for getting Rubik's name right!