Kiosk Mode is a setting on the OS (Operating System) level that actually locks the device (PC, mobile, tablet) from running any application apart from the applications that have been set to run. Usually, it is one application that serves one purpose and being used as a local dedicated service for users, using touch screen capabilities – hence the name Kiosk Mode.
The Kiosk Application is an application that runs on the device level, without any OS level intervention, and allows users to perform a task (getting information, downloading coupon, giving service feedback, asking for specific service, location and way finding and numerous of other potential applications), using the touch screen capabilities (also kiosk – this is the main cause for confusion).
Digital Signage software implementers often uses Kiosk application of many kinds to provide location/time based service to visitors at almost any kind of venue. It would be also correct to say that most likely, the Kiosk Application will run over a Kiosk Mode setting – as it is the safest way to prevent miss-use/damage/abuse of the digital signage kiosk.