I agree with the original question in that the two terms are interchanged quite often. The curriculum refers to what a teacher or instructor must teach. It involves the identification of specific concepts or ideas or techniques that must be revealed and taught to students over the course of a specific term. The syllabus is a more segmented approach on how these particular concepts will be delivered. The curriculum refers to the what, and the syllabus is more focused on how and when. Both are needed because the curriculum offers the overall goal and the syllabus proposes how to get there and when that arrival will be anticipated.
A syllabus is typically a schedule of curriculum for a given school term. So on the first day of class, I hand out a syllabus, which talks about the expectations of the course, guidelines for grading, and a schedule for what material, assignments and tests will take place during the quarter.
A syllabus, then, includes curriculum, among other things. Curriculum is the content of the course itself. The chosen concepts and ideas, the core material you expect students to learn. A syllabus also typically includes the materials such as books, articles, websites, etc. that a student is expected to use to learn the curriculum.
The term curriculum is used more to describe the broad range of subjects included within a course of study, and syllabus refers to more detailed description within each of the subject.
Please not that there are no hard and fast rules that clearly define the degree of detail associated with either curriculum or syllabus.
For example a curriculum may contain just a list of subjects such as physics, chemistry, and maths; or may give more details of topics within each subject. Thus instead of just saying mathematics it might give more detailed list of topics like co-ordinate geometry and within that further details like parabola, ellipse, hyperbola and planes.
Syllabus would generally contain much more details like: general equation of a plane, direction ratios of the normal to the plane. distance of a point from a plane, and angles between two planes.