It all depends on where and when it is. A good answer for the U.S. may not be the best for the U.K. and Germany.
MEs can be used to replace many engineering specialties. They are generally more versatile. These MEs are also used more frequently in managerial roles within corporations. EEs, on the other hand can deviate from their careers and go into programming or software engineering.
ME is more suited to people who have a good understanding of the physical world, such as physics, design and architecture, civil engineering, and others. EE is better suited for abstract thinkers (such as math, chemistry and programming). These are minor differences. If you’re able to be an engineer you’ll be a good at both.
All things considered, the best engineers are those who have either a Bachelor’s or Master’s degree in engineering. Engineering is about having a solid scientific foundation and current knowledge in your chosen field.
Some comments The same question was asked of me in high school. My first choice was EE, as it was more prestigious then. But, failing to choose that, I chose ME. I have never looked back, and I believe ME is much more appropriate to me. Ken Masterson has the answer. Find your passion and follow it.