I am a MMM form Kellogg, and when applying also considered the MIT LGO. So let's be clear with the differences.
1. Career Focus: If what you like is manufacturing, or core operations, your first choice should be MIT. It is much stronger on that side than McCormick. The downside is that it is too distant from the business side, Sloan. People in the LGO hardly interact with their fellow Sloan classmates. On the other hand, MMM's are part of Kellogg, we share the same classes, activities, etc. We have MMMs recruiting for IB, Consulting, Marketing, as well as people going to Design and Ops.
2. Timeline/Internship: LGO offers you a guaranteed internship. That has pros and cons. The pro is that if your interest is within the partner companies, you nailed it. But what if you realize that maybe hard ops is not your thing and want to try something else? Then you might want to have more options. After all, everyone gets internships in all tops schools.
3. International Experience: LGO is for US residents. That is how the program was created, and the purpose behind the partners. It has 3-4 international students per class. You loose a lot of experiences with that restriction. The MMM shares the profile with Kellogg, ~30-40% of internationals.
Summary: Both are incredible programs. If you are convinced you want to go to manufacturing/operations, don't hesitate, LGO is your #1 choice. If you want to have an MBA experience, while strengthening your Ops and Design knowledge, without leaving aside an opportunity to pursue a career in the business world, the Kellogg MMM is what you need.