First off, this post is really about the young ones (18-22), who work on an undergrad degree. Post-grad degrees are really different, and don't serve the same purpose. PhD's are even more different, and aren't even degrees anyway, you've probably got a few by then.
My first point is that I think that young people like that are leaving home for the first time for the majority of them, and also have not been pushed to work on their own initiative. They also haven't had the opportunity to learn the skills that an undergrad degree can give you. Being at university gives them the support they need, both emotional and practical, and it also allows them to mix with people their own age, and learn that drinking 12 pints will make you sick.
On a degree program they learn to organise their work on their own, write at length will full references, which means they have to do substantial research on a given topic, learn things quickly (modules can last just a term), learn to present effectively, work on a team assignment (which seriously is the biggest issue usually), there are workshops on interview skills,...there's lots of benefits.
You can learn these things from going straight into industry but you learn in a different way. You don't pick up the skills you do at uni, you pick up others, but when undergrads make their way into the industry, they also learn these skills. Admittedly for most of them the transition from Uni to the work place is a bit of a shock at first. The advantage that I think they have is that coming from degrees like marketing, computing and business for example, they are able to put SEO into context, because they are familiar with the wider picture. That is really useful. They can also provide a different perspective which can only be useful to an employer.
The limitations of a degree is of course first off the cost, and then the fact that in Uni you don't learn many professional skills unless you do a degree which is vocational in nature. Those related to SEO tend to teach you useful skills though. I think that working 9-5 is usually a shock, and then finding out that it's actually 9-7 is a bit worse. Working in an environment where it can be noisy and being asked to do lots of things at the same time is hard too.
Actually one guy said to me "I'm so tired and its only Wednesday, I don't know if I can keep it up 5 days a week" - He did though and he's doing really really well now.
Having a degree in SEO however would just negate all the benefits of coming from a different background. You only learn SEO, maybe a module in web design or something, but you wouldn't learn about information retrieval in any depth or business, or if you did I suspect it wouldn't be a very long module. It's a shame in my opinion to not take advantage of learning something else to put SEO into context. Also, I think it's best learnt on the job.
You can also take another route which is to work in SEO and study at the same time. Some people shudder at the thought, but it's a very efficient way to get the best of both worlds. It is possible, I'm doing it and a lot of others are doing it too. It's hard work, but you get used to it and it's rewarding.
I came in the SEO profession by chance really. I spent a lot of time building search engines, pulling Google apart, building indices, becoming proficient in NLP and one day someone said, "oh could you optimise my site?" - I looked a bit puzzled and said yes. That was my first project. I quickly figured out that I had a lot to learn, and so I spent lots of time learning everything. My background in linguistics and computing helped a lot, and it still does. SEO was a bit different back then but you move with the times and that's exciting.
Those coming into the profession at an older age can draw from their wealth of professional experience, which I guess is the same as learning all about a different subject to SEO, and bringing with them an awful lot of knowledge colleagues can draw from and enjoy.
A masters is useful if you want a career change - I went from a degree in translation and linguistics to a masters in computing (machine translation). A PhD is worth doing if you want to work for the big search engines, in the cool labs, in academia and mostly consider doing it if someone else is paying!
Coming from such an academic background, I am obviously going to favour having a degree. If I didn't think University was useful, I wouldn't have stayed 10 years.