Both majors will give you the opportunity to enter the construction business, but the construction management major will focus more on the business aspects of construction, while civil engineering will focus more on design. There are plenty of people from each background in this field (architecture majors also make up a significant portion of the field), so choosing one over the other won't put you at a disadvantage.
Personally, I think going with civil engineering is best if you can handle the engineering aspect of it. It just leaves more options open if you decide in four years that construction isn't for you. You can jump into the design side without any problems if you have an engineering degree, but if you have a degree in construction management, that may be a problem because design firms typically like to hire people either with or have the ability to obtain licensure.
In terms of what you learn, in the civil engineering curriculum, most schools allow you to take a few courses in construction. The number varies from school to school, so check with yours. I saw a few curricula in construction management, and there are quite a few similarities in coursework. If you end up choosing the "wrong" major, it probably won't affect you much anyway as long as you can keep up with the work.
This isn't really a decision to stress out about if you're absolutely certain you want to go into construction. If you're already enrolled at a college, take a look at the degree requirements for each major and see what you think you would like most.