Well, I'm not sure if natural ability comes into it much (it's not mentioned in any of the dictionary definitions to which I've referred).
The difference, if there is a huge one, seems to me to be more one of
having the training or skill to do a job well (skilled) "versus"
doing something very well (because of training or experience, not that this training or experience is the focus. Differing construals of the "same" thing, again?*). The
Longman Language Activator was especially clear and helpful in its definitions:
[adjective] someone who is
skilled at a particular job
has the training and skill to do it well
British /skillful American /ˈskɪlfl/ [adjective] someone who is
skilful does something very well because they have had a lot of training or experience
Cambridge Advanced Learner's: skilled -
having the abilities needed to do an activity or job well
skilful -
good at doing something, especially because you have practised doing it
COBUILD3: Someone who is skilled
has the knowledge and ability to do something well.
Someone who is skilful at something
does it very well.
Now of course, it is well-known that lexicographers consult each other's work (more to ensure at least as full and comprehensive a coverage as the competitors than to plagiarize!), and this might have resulted in one, original analysis of meaning being used as the basis for all subsequent and similar ones, but there is surely something intuitive about all this (and doubtless also to do with a repeated and thorough examination of multiple sets of evidence too).
Anyway, what might make things clearer is a bit of paraphrastic wizardry:
Can they
operate (="Are they skilled at operating") the machinery?
vs.
Are they
good at operating (="Are they skilful at operating") the machinery?
This kind of paraphrase avoids having to make the "choice" between "skilled at" and "skilful at".
Interestingly, whilst "skilled" is the more productive of the two words (entering into phrases such as "semi-skilled"), and is given more coverage in dictionaries, it is "skilful" that seems to be the "jucier" item in a thesaurus.
* On "construal", see my first post in the "Relative pronouns" thread.