The problem you have is they have the money to through at shutting you down but do you have the money to try and prevent it?
We had a similar case with our business name conflicting another global name, they tried to bring it under trademark infringement and went on for weeks. We came to a suitable conclusion in the end though.
If you where using the name and logo before they were about you may have an argument but it will cost a fortune to argue it.
Approach them for a free licence to keep your current name/logo without restrictions in it's current form, that will cover them for any possible future infringements and will stop anyone being able to use you as an example of infringement as they can just say you are under licence and refuse to issue any future licence's.
At the end of the day it is such a different market no one could ever confuse the two of you so they are just being pathetic.