To me, skills are something related to mechanical performance. Someone is very skilled at playing football, for instance.
Competence/competency on the other hand is more related to knowledge. Something that is not mechanical.
What is the correct use in this very specific scenario?
Suppose you have the Stackoverflow site (asking questions and answering them). Which word would we use if we wanted to describe individuals with the highest reputation in specific tag? Like C# (as in C# language). Do these users have competence/competency or skills?
We could say He's very skilled in C# language, but in a differently formulated sentence, it would be better to say His competencies are (in) C# language, software algorithms and ... Or would it be better if we used skills? My gut feeling says the former is better. But is it really? And why?
And while we're talking about it, why not also define the difference between competence and competency?
I would also like to add the third term that could be used in this specific Stackoverflow scenario. Proficiencies. I suppose this term could be used only for those users with gold/silver badged tags (similar to what users have here if you don't know Stackoverflow). Or describing users with top reputation in general without explicitly listing their top ranking tags.
In such scenario we could also talk about experts with expertise (is there even plural of this word?).