I was self-taught. I started out in jr. high school on 30cps clacky terminals dialed into M.E.C.C. (anybody else in here know what that acronym expands to?)
But then I went on to get a CS degree with EE "concentration" (kinda like a minor but not as much work). The EE work was not trivial- I took 4-level courses on things like signal processing, vsld, semi-conductors, etc. etc.
As a result, I graduated knowing three things:
1. What a computer can DO
2. HOW it does what it can do
3. How to MAKE it do what it can do.
In other words, I understood computers soup to nuts (or thought I did- I still had a lot to learn). When diagnosing a problem or architecting a solution, I think holistically. The phrase I've been frequently accused of over-using is "Silicon to Glass," meaning from the silicon in the chips all the way to the glass screen of the computer monitor and everything in between.
To an employer, this probably doesn't mean squat. They're looking for Skill XYZ. And when they hire you for Skill XYZ, they really have no intention of using you for anything else for the entire time you are with them.
To me, it means everything because while I'm working for an employer and utilizing Skill XYZ, I'm also looking for opportunities to learn Skill ABC and apply it to my current responsibilities. And then Skill ABC goes on my resume.
As a result, my resume looks impossibly broad, with real, working, got-paid-for experience in a diverse range of disciplines, from large-scale (many thousands of nodes) network design, telecommunications, database architecture and application design (I've designed systems that earn $100M/year). Not only that, I've spread out vertically as well, working in as many industries as technologies.
The thing I ALWAYS credit is my CS degree. Without that intimate understanding of what's going on inside the systems and software that I create and use, I would be simply (as another poster put it) responding to interfaces, not utilizing skills.
What freaks me out is how a large majority of my co-workers are one-trick ponies. They know how to code Informatica data integration mappings. Or they know how to write Perl scripts. Or they know how to create SQL Server databases and monitor their performance. Maybe they have a minor secondary skill, but that's usually it. I always ask that type of person if they have a CS degree- I've never had one reply "yes." Turn that around, and when I find that a co-worker has a CS degree, it doesn't really matter what we originally hired him or her for- if a job needs doing, that person will either apply existing knowledge to the problem or immediately go about acquiring the required knowledge from whatever sources are available- and if nothing exists at the time, they will CREATE the tool that solves the problem. Because a CS degree is just that: a set of "tools in the tool belt" that can be taken out at need- and some of those tools are designed specifically to create other tools. Self-taught folk are fine, but I've never found one with the breadth and depth of understanding that you get even from a newly-minted CS grad.
When I'm hiring, I'll take a CS grad with diploma still dripping ink over a "expert" in some tool or technology ANY day. Because the former has demonstrated the capability of picking up any tool and applying it (or making his own), but the latter has only shown the ability to use one.