Most companies use the title "Software Engineer" and "Computer Programmer" to mean the same thing. It's basically programming and not
hardware work. Never heard of anyone with the job title of 'software programmer' being used in the States.
Not to mention "Software Engineering" is a common undergraduate major, and largely focused on programming. Almost same as "Computer
Science". However, "Computer Engineering" tends to be more hardware based.
Anyway, in my case, I'm finishing a five-years degree of Computer Engineering. I have knowledge in Maths, Physics, a bit of electronic, hardware, programing and also Software Engineering. So, will my collegue education be relevant to my career choice? It will be quite relevant in my case, but certainly I can end up being many things and I'm not sure what my future holds next. I just hope not to be sitting next to a High School diploma guy doing basic programming, but that of course will depend, as has been said before, on the each individual's skills, the title means nothing but a prove of what you're able to do. So, if you can, choose the degree you think you will be good at, if not, don't give up, just show the world what are you capable of and don't worry if your future career has nothing to do with your previous education.