I can respond from a perspective shaped by U.S. trademark/service mark law only. Very broadly: Both can exist with similar names if that's OK with both companies. Often if you're operating in different industries, both parties will be fine with it (Sysco and Cisco).
If the other firm has trademarked the name and your firm hasn't, and you move forward with using the similar name in a similar industry without making the other company aware of it/asking for their written OK, in the U.S. you would be making yourselves vulnerable to a lawsuit.
Think of it as a shared driveway... I don't know if this metaphor will translate culturally! But: Your house and the house next door are separated by a single driveway. The land belongs to the house next door, but your neighbors never, ever use the driveway and looking at it from the street, it might belong to either house.
Would you make a practise of using the driveway without asking the people next door? If you did, how might they respond? If the driveway belonged to you and the people next door started using it, how would you respond and why?
Try to think it through like that. Hope it helps, good luck!