What is the difference between Masters in Engineering Management vs. MBA?
I haven't heard much of an MEM before, but it sounds like a quirky, industry-specific degree, which I think is the key here. An MBA is more established, general in scope, and applicable for management careers outside engineering. So recruiters would know what an MBA entails without needing to ask. Not so much with an MEM. 

To me, the question isn't which to get, but whether you should be getting a management degree or not. The traditional advice for getting an MBA is don't, if you're a fresh grad or new worker/hire. From your question, it sounds like you fall into one of these categories. 

Typically, the merits of having an MBA for job recruitment are vertical, not horizontal. It's key to moving up within a company, once you're in one, but it doesn't improve your chances of getting hired off the job market a great deal. 
So the time you should consider getting an MBA is when you've been with a company for a while, and within the company, are ready for a promotion. In some cases, companies would even fund your MBA studies at this point.