I have worked in both public and private sector over the past 15 years, everything from SME employing 50 people to big multi-national company or local council employing tens of thousands. The biggest difference I've found, in relation to skills, in comparing working in the public and private sectors is one of variety and career path.
I've found that in the private sector, in all but the smallest of companies, you are usually quite restricted in the work that you do and the skills you are expected to apply. You are also likely to have a fairly clear career path in terms of knowing where you are and what skills you need to develop to move to the next level up.
In the public sector career paths are less clear and you are typically expected to pick up a broad range of tasks and skills quite quickly and to a fairly high level. many of the consultants I've worked with have been surprised at how myself and my colleagues have switched around roles with a project and between different projects.
I think the driver here is around the restrictions on hiring. In the private sector if you know you need someone with a particular skill set for 6 months then you'll probably negotiate funding from your manager for a temporary contract, hire someone through an agency and that's it. In the public sector it can take 3-4 months to get approval for a temporary contract, another 3-4 to have the job description agreed and checked by HR and legal then 2-3 months to do the actual hiring as even if you go through an agency you usually have to advertise publicly as well for equalities reasons. Usually much quicker and easier to just grab an existing member of staff and tell them that it's now their responsibility, if they don't know how they should just look it up on the web (in their own time) or buy a book (at their own expense).
If public sector workers have difficulty explaining their skills it's probably because they have had to develop, use and discard such a variety that it's more of a question of which skills they should be talking about.
Really the biggest skill a public sector worker has is adaptability!