Software engineering student at SJSU here, and I wish I’d started with computer science. When it comes down to it, the difference between the two majors is pretty much negligible (which is why I’m not bothering to switch), but SE seems to focus more on the practical side of software development while CS focuses more on the theoretical. As for why I regret taking the former, the simple fact is that I feel like CS is more what jobs are going to expect you to have learned from your degree—you can pick up (and will probably have to anyways) a lot of the nuances of practical software development on the job, while the theoretical side is probably better to be taught in a university environment. There’s also the fact that the theoretical side just plain interests me more.
In the end, a CS/SE degree is always going to involve a lot of independent study to be at all complete anyways, since the subject is so general and varied that it’s not likely to ever go at all deep into any specifics that you want/need from it.
tl;dr, if you’re more interested in the process and the software development life cycle, you’re probably better off going for SE. If you’re more interested in the theoretical stuff (data structures and algorithms, operating systems, compilers, language theory, etc.), go for CS. As for what the degree gets you in terms of resume, I’m pretty confident the two are almost identical.