I have some fair knowledge about the MiM at LBS and LSE, so i can give my 2 cents about them. Also both the schools are pretty well known & are excellent choices, so you cant go wrong with either.
These are the differences according to what ive found:
1) LBS is a fast paced 1 year program while LSE is a more theoretical 2 year program. In fact LSE describes its MiM program as "unshamedly academic" and touts it as a program to build a sound business foundation.
2) LBS costs around around 27500 pounds while LSE is a total of almost 50000 pounds, of course it is a 1 year vs 2 year program
3) LBS core values are more towards leadership and dynamism while LSE preaches a more in depth critical thinking approach as can be observed via its core value "Analyzing the cause of things"
4) Base Salary of 36,295PS & 97% accepted a Job offer within 3 months of graduation for LBS
86% in work/further study 6 months after graduation & 38,100 pounds avg salary for LSE
So in conclusion if you are looking for a faster way to get a good business education and get into the industry LBS is a good choice, while If you want to build a strong business foundation and dont mind a more in-depth 2 year course LSE would be optimal.
As for getting into consulting or IB, don't worry. All top consulting & finance companies recruit from both LSE and LBS, its a really great way to enter both these sectors.
Hope the answer helped.
You can check out
https://www.mim-essay.com/lbs-mim-review.html for more detailed review about the LBS MiM, as well as info on how to crack their admission process.