It depends on the degree of similarity and the overlap between the goods and services each business offers.
If the existing trademark covers a completely different class of goods or services, the risk of conflict is low and you may be able to proceed with your own application without significant issues. Two identical names in unrelated industries can coexist, provided consumers are unlikely to confuse the two sources.
If the existing trademark operates in the same or a closely related class, the risk is higher. The owner has legal grounds to oppose your application and, if successful in commerce, to demand you stop using the name. In that situation, the options are to negotiate a co-existence agreement, adjust the mark, or rebrand.
A trademark search before investing further in the name will give you a clearer picture of the risk. Get a free lawyer's check with Trama to assess your options.