It is best not to talk in generalities regarding these areas of study, as their interpretation will vary greatly from person to person, and from department to department.
So from the Drexel programs, here are the differences in the textual descriptions, and other high-level info from next year's course catalog:
MSSE is in the College of Information Science and Engineering, but co-sponsored by the College of Engineering
MSCS is in the College of Engineering
This is unfortunate if you wanted to double major, as you'd have to be dealing w/ 2 separate colleges, but you'll be dealing to some degree w/ the College of Engineering w/ either.
As for the high-level descriptions of the programs, the gist is that SE is about practical applications, with the chance to deal with broader subject areas through the "track" system (one of these is CS, which would draw this much closer to the CS degree). The CS degree is focused on some practice, but also theory and the research interests of the faculty. This prepares you more for further research or academic pursuit than the SE degree might. What really codifies these differences is the stress, though it is not required, on the Co-op program for SE compared to the stress, though it is not required, on a thesis in the CS program.
Between the CS and Engineering track of the MSSE, there are 27 courses that are mentioned specifically in the catalog that are also mentioned specifically for MSCS. The track most divorced from CS is the InfoSci track, which shares no specific courses with CS.
There are only about a dozen CS courses that are mentioned specifically for the MSCS program that are not mentioned for the MSSE program. Again, if one were to choose the InfoSci track for the MSSE, there are many distinct courses from the MSCS. There are quite a few for the engineering track also, only specifically sharing 3 ECEC courses. Sensibly, for the CS track there are only 2-3 courses mentioned in the MSSE that are not mentioned in the MSCS, but these could still be taken for the CS degree.
It seems like the best compromise would be the MSSE with the CS track, but breadth may not be what you are looking for. You didn't say either way, but I'd deduce that academic research is not a long term goal, but there are other avenues that the MSCS offer that may be more appealing to you than MSSE.
The long and short of it is that what we, other members of industry, the university, the respective colleges, and others think of the two tracks is largely irrelevant. What matters is what you (and your employer, since they're your benefactor in this endeavor) want to get out of it. My opinion of the matter would be to go CS, but I'm jaded because that's what I have my (undergraduate, and only) degree in. It seems that SE is a younger program, that is an off-shoot of CS designed to prepare students for a career in industry. CS provides this as well, but stresses research and future academic pursuits to a greater degree.
Have you contacted advisors from either college to get their input on this matter? I'm sure this is a pretty common question from perspective students that they probably have some insight on.
Either way, good for you. I'm jealous that you have the opportunity, especially on someone else's dime (though I'm guessing their buy-in gets them some commitment of future service from you).
-Lee
EDIT:
Responding specifically to:
Computer science is a degree in which one extensively learns about software and hardware and works "more" in a company's research division rather than in a development division. Software engineering is a degree in which one extensively learns about software with potential electives in hardware and works more in development, rather than in research. Is that somewhat close?
Unfortunately, it is really not that clear cut. Certainly the MSCS at Drexel focuses more on research, so if that was the sort of position you were wanting it might be more applicable. However, you can get plenty of hardware background from the MSSE, it isn't limited to those pursuing an MSCS.