I do not think there is a universal rule to distinguish between the two master degrees you described.
When I attended the University of Florida for EE in 2001-2002, two master degrees were offered: Master of Science and Master of Engineering. At that time, the only difference was this: If your undergrad degree was in engineering, then you could choose to receiver either the M.S. or the M.E. Otherwise (e.g., your undergrad was in physics), you could only receive the M.S. The course, thesis vs. non-thesis, exit exam, etc. requirements were exactly the same for both M.S. and M.E.
Since just a plain old M.S. degree is widely known, I (and all my classmates that I know of) chose the M.S.
I think I've seen other universities distinguish between the two flavors by thesis (M.S.) vs. non-thesis (M.E. or M.S.E.), with the non-thesis option replacing the thesis with more coursework or coursework plus a smaller project.
There shouldn't be a difference in job hunting just based on the name of those two degrees. As somebody who occasionally conducts interviews, I would not care. Of course, if for the program that you're considering, one degree implies means more effort than the other (e.g., one requires a thesis), and you use that to sell yourself, then it (the extra effort, not the name of the degree) will matter.