Computer science is defined as the broad field that encompasses but is not limited to: the study of algorithms, programming languages, computer hardware and software, networking, operating systems and architecture, and ethics surrounding the entire field. A decent computer science degree will try to include all these topics.
Engineering is the application of mathematics, science and other fields of knowledge to produce cost effective solutions to mans needs. Engineering students study more vocationally oriented things, and there is a large overlap between computer science and some specific engineering degrees.
What field of engineering do you want to study? It would also depend on what you actually want to do, no?
Mechanical engineering and aerospace or automotive are closely related.
Electrical engineering and electronics is closer to computer science overall. Chemical and process engineering have some overlap.
Industrial and production engineering are the newer forms of engineering (though they have been around for a while) and are more management and business process oriented. Financially I would not know which is better, but work experience and skills + good grades, university reputation and connections. are some determining factors.
Also please note that you will see non cs degree and non engineer degree holders working in traditionally cs and engineering degree holder positions. Work experience and skills can be developed and honed over time, so don't worry too much about this for now, focus on what you feel suits you better.