Square is for small or entry level merchants who may be in their first year and can't afford an actual production grade POS or web integration, stripe is for those who've realized they've reached that point or have encountered Square's limits, and by them moving into the entry level market by supporting this method of transacting with iDevices, I'd say square should be considering their next move.
Eventually they'll just be an app that runs on the phone, making it significantly easier for a business to signup to use their service.