I'll chime in here with a non-expert opinion. I'd say that the etymology of these two words is also very important to consider in the discussion. There is, of course, a long list of English adjectives that could be used to translate these words based on context, not to mention the subtle nuances and differences between them (which can also get a bit tricky given that nuance varies between dialects, regions, individual speakers, etc.).

However, (in my opinion) a very important aspect to consider is where the words derive from. In English, the two most obvious translations for "blyg" and "skygg" are "shy" and "timid". The words are very close, but the biggest differentiator is that "timid" derives from Latin (timidus) and "shy" derives from some old German root (like Old English, scēoh or German "scheuen").

In a very crude sense, timid is simply the Latin version and shy is the Germanic version of the same concept.

I suspect the same could be said for blyg and skygg. Certainly, the two words have subtle differences, but I suspect "skygg" shares the same etymological history as English's "shy" and German's "scheuen" (BTW English and German are classified as West Germanic languages), whereas "blyg" appears to have its roots in Old Norse (i.e. North Germanic).