here my take.

cloud app = app on mobile phone, 5MB installation Android, one can sign in with FREE plan/account (FREE plan restrictions apply like computation memory & computation time, file management), full access to Wolfram Language, you can code while you're on the bus. you can upload .nb .m files from your PC to your FREE cloud account and open and edit/save them thru the cloud app. If all you need on the go is a FREE mobile coding environment without file management, you could just use the sandbox instead. for all mobile coding, an internet connection is required. it is a must-have app for us users and in theory also replaces the need for the NON-FREE alpha app because one can invoke alpha queries from the notebook environment, right?

player app = app on hdd, 1GB download. huge installation on Windows. Also FREE and more like a viewing/viewer software. Not suitable for creating code. it's an app mostly for people who DO NOT have a full installation of Mathematica. I liked it for test installation and then uninstallation :D