However, there are differences, and the specific focus on certain areas is good if you eventually want to go into that specific area.
My advice is that if you know you want to go into a specific area already, then you're slightly better off choosing a course specific to that area; but if you're not quite sure which area of chemistry you want to go into, you're probably better off with a general course, which might help guide you towards an area you enjoy for the future.
Chemical Engineering entails a lot of math, especially in alternate coordinate systems (heat flow out of a pipe in cylindrical coordinates, for example), and a lot of empiricism (we don't know exactly why this is, but it works) rather than pure solutions from first principles. It has intense math, but not empiricism, in common with physical chemistry.
In the United States, you usually only narrow down your focus more specifically than just "chemistry" at the post-graduate (masters and doctorate) levels. Where that's not the case, you are usually preparing for a very specific field, like polymer chemistry. That usually helps your job prospects at graduation (these programs are usually created to fill a need) but limits your flexibility down the road.