Remember, there's an inherent goal behind their app and yours. Within your markets (as niche as it may be) they may have a different vision and that will make a difference further down the line.

Within a year, nobody is going to look back and call you a copycat -- unless your goal is specifically to build an identical version of that they're building.

Overall, don't worry about it. Polish yours and get ready for launch. You want to find differences? That's what features/support/pricing/service/design is for.