What should I do if my business name is similar to another business?
It is the job of the Massachusetts Secretary of State (or the Secretary of State of any other state, for that matter), to make sure that the names of new business entities are "distinguishable" from any previously registered business entities.
In other words, they can't conflict. For example, if there were an �XYZ Corp.� already filed, you could not register �XYZ LLC.� In your case, there would seem to be a sufficient distinction between the two names ("North Shore Computer Tutor" and "The Computer Tutor")--assuming that there are no other company names or registrations that would conflict.
However, to be on the safe side, you should also plan to come up with alternative company names in the event that your name can't be registered. Because under the law of your state, some business names are subject to additional restrictions, you'll want to speak to a local attorney to help you through the process.