(from forum member AlienX) The answer is not easy. Though I can try to explain it in simplest terms possible.
(To understand what I am saying, you need to empty your cup. Let go of all opinions, whatever people say, general impressions, your ideas etc.)
CS (Computing Science) is scientific level discipline. A problem solving career. You'll do research, write research papers and present solutions to known or unknown problems. The beauty of this career is it being open-ended. You can work in engineering (if you want to), nano technology, quantum computing, solving centuries old computation problems and so on, imagination is your limit. You must note that in CS you do not care about implementation.
Alan Turing, the person behind modern computing was Computer Scientist. He is founder of computer science, look him up. Your mobile, PC, Console, Digital watches, iPod, iPad, everything is based on his theory.
SE (Software Engineering) is exactly what it says. You will be implementing the solutions, develop software, code and basically follow the guidelines computer scientists have presented. SE is wholly focused on software. Don't get fooled by the word 'engineering'. Engineering doesn't mean you'll be dealing with hardware. In fact, you deal with hardware more in CS in comparison to SE. SE is a sub-branch, or to be precise, a specialization of CS. Most universities don't even offer SE as a separate degree, it's mostly a specialization you can choose in CS. It involved code quality, teamworking etc.
What you want to be depends what kind of career you like, if you are more theoretical person, a problem solver, have innovative ideas, and like to research a lot, CS is best for you, Algorithms, Data structures, Compilers, Language creation etc are all CS related. If you ar;e more practical minded, love to create software, code, or program robots etc, SE is best for you.
For example, Intel processors work on x86 architecture, which was presented by CS people, in the form of research papers, technical documents etc. How it came into the shape you are using was Engineering job as how to code it to work with registers, cache and how to implement it as millions of transistors etc.
Computer Science > Software Engineering.
Don't get caught up in opinions, people base their opinions on rumors rather than facts. I am sure when guests visit you and you say engineering they cheer up and when you say computer science, they will think you were nalaiq and got into lesser career. Well we as a nation are dumb people, literate jahils. This kind of ranking careers on the basis of proving yourself to be superior in your family/friends has brought us to the point where people like amir liaqat can shove mangoes in your mouth.
If you like to code your own programs, love to program robots, Like practical solutions than Software Engineering is way to go. If you like to create algorithms, create new languages, design solutions, research a lot than Computing Sciences is way to go.
Have fun .