Js55 Basically nailed it. I'm graduating shortly with my bachelors in Welding engineering, there's only about 4-5 schools in the nation that have welding engineering programs. That being said the job market for a welding engineer is good right now.
Also out of school I would feel confident in my ability to pass the CWI test with studying. My particular school has 2 classes on Non destructive testing so I already have experience and education operating UT Xray, Mag particle, etc and I've done quite a bit of work with the AWS D1.1 code book (as I type this my hand is resting on D1.3 and D1.6 . I hope it doesn't sound like I'm bragging, but I'm glad my school gave me a taste of QC work if I should ever choose to go that path.
The careers are very different in terms of both job description, training, etc. Js55 explained it better than I could.
there are two types of welding engineers. The extremely rare few who actually get a degree in welding engineering; and Mechanical engineers, metallurgists or occasionally skilled welders who work their way up to the work without a relevant degree. ( I know an extrmely skilled welder going for his bachelors in buisness so he can become a welding manager/engineer). Welding engineers generally tackle "engineering" issues ranging from metallurgy like picking the rod type and determining metal strength. To design like determining weld size or type of process to use, and finally management like overseeing welders, developing weld procedures, testing etc.
CWI's and QC exist on the other side of the fabrication world ensuring that work was done to code and properly and testing welds, they are the check valve to make sure sub standard products don't go out the door. It requires a different set of skills than a WE and is generally field work.
My suggestion personally would be become a welding engineer, you can always pick up a CWI cert later I know a few WE who have a cert for freelance and consulting work, but often they remain in the realm of a engineer.
what's right for you might not be what my opinion is. Some people don't want to take 4 years of college and I understand that I was one of them. Right now the job market is good so theres work in any welding related field. Hope that helps, I love the Welding engineering program and enjoyed my internship.