Engineering vs Science degree?

In terms of job qualifications, how similar are engineering and science degrees? For example, would a Chemistry major be qualified (if disadvantaged) to accept an entry level chemical engineering position? And conversely, would a Chemical Engineer be fully considered to apply for a strictly lab science or statistical modeling job?

I have heard that engineers are qualified to hold jobs in engineering and otherwise, but straight science majors would not be able to hold engineering positions. This would mean that as far as job options go, there is no reason to be a science major (save for immediately pursuing graduate school).

I KNOW I want to do something in the environmental field, but one college that I absolutely love would force me to follow the science route, and my second choice has both engineering and science majors. As far as a career goes, I have no idea whether I would prefer engineering or research/lab/education jobs.

BLS lists scientists as having more occupations and openings, yet my parents are telling me that engineering is the way to go to have a shot at actually getting a job.
http://www.bls.gov/ooh/architecture-and-engineering/environmental-engineers.htm#tab-6
http://www.bls.gov/ooh/life-physical-and-social-science/environmental-scientists-and-specialists.htm#tab-4

Thanks for any insight or examples.

You can get jobs with a science degree and you can compete for entry level engineering jobs, particularly with a physics degree. Chemistry is somewhat different since Chemical Engineering does not have all that much chemistry in it if you look at the typical curriculum. I would say that most chemistry majors will find jobs as analytical chemistry technicians. This is an area that always needs people. Engineering generally is more directly applicable towards a job after college. Ultimately though, the job that you find as a science major will depend on how well you sell yourself, your experience (internships and so on) and the contacts you make during your years in college.