The Computer Engineering degree would most likely get you a job that earns more money. It would set you up with a crazy amount of programming knowledge.
Website and Design might be more flexible. My wife does this and works from home, that is what makes me say that. It does not involve heavy programming. It does require excellent design skills (which you will NEVER get in a computer engineering program) and people skills (also somewhat lacking in engineering programs).
So, similar but different. I don't think either one is better or worse. Go with what feels best to you and fits your personality and future goals.