USA Eligibility

This varies from one school to another. In terms of general understanding, an MS tends to require 30 graduate credits along with a thesis or rarely an applied project. Typically, this is aimed by candidates intending to further work on a doctorate or with keen interest in research work.
On the contrary ME /M.Eng tends to be focused in meeting needs of engineering intending to further broaden their knowledge and tends to be often a mixture of engineering and business class. As earlier stated, this definition does not necessarily apply to all schools and need to be verified through the respective department webpage.
M.Eng in most of the colleges are supposed to be for students pursuing the degree course with prospects for making a career in the industrial line. For example: getting a job after graduation. If your intentions are to get a job, then M.Eng is best for you.
However, if you are research oriented and would like to conduct researches or maybe go for Ph.D after graduation, then you would be faring well in MS. Since, Masters course are usually thesis and non-thesis oriented. You need to choose according to your academic and career plans. M.Eng is the non-thesis option and MS is the thesis option in most universities. So, both are better than each other in their own context.