I have an app in the store that's been doing reasonably well. Another developer released an app with the exact same name. (I'm not sure how, as the App Store normally prevents that). I filed a complaint with Apple, and they renamed their product, simply adding "Pro" to the name.

Search for any popular app name and you'll see the other apps trying to ride off the same search terms. Apple seems to be fine with that.

Where Apple will get involved is if you send them a properly worded legal request based on an actual trademark or intellectual property violation. They will act if they are compelled to, but only then.