The two illnesses are the same. One you can test for via Labs, the others show Lesions on the brain.
What I see often is the opposite of what you ask. Many MS patients find out they have Lyme and not MS. Those with Lyme who do have brain lesions and treat the Lyme, have their Lesions go away.
There are hundreds of links on this on Google, it's not coming from my opinion. It another one of those things that take time to settle in.