software engineers can be programmers, but still, not strictly.. within the development of a software, you usually have product owners, developers (coders), testers, architects, infrastructure analysts, UX and others.
software engineers are usually on top of it all, not to be confused with project manager, albeit often they carry this burden too. they are the ones that have to understand how everything will work, from the objective of the product as to how it can be escalated if needed, contiuous development, methodologies, testing, support, etc.
Imagine assigning a mechanical engineer to be your car mechanic... that's how you can 'compare' the software engineer with a programmer.
Please note that this may vary from company to company and more often than not, people hired for IT hardly work exclusevly on what they graduated. Also, I might be stepping on some dev's toe by saying that they are not software engineers :D