The PlayStation App provides a few core functionalities that improve the gaming experience. They can browse the PlayStation Store, make purchases of games and add-ons from afar, but also start downloads directly to their consoles, quite helpful if you want your game waiting for when you get home. It also gives players the ability to view trophy collections, which track their in-game accomplishments and compare progress with friends.
Recent versions of the app have brought substantial upgrades to voice chat and messaging. The newest update allows players to join into party chats from within the app itself, so friends can be talking while you are no where near your console. The system for viewing and managing friend lists, sending game invites is easy to use as well.
Still, the app is flawed. Other users have also experienced some connectivity errors, mainly trying to remotely connect the console after putting it in rest mode. The second-screen capability, a major requirement of the original model (and up until fairly recently one you predominantly hoped not to use with suitably future-connected games) has been drawn back in recent updates. The app also supports game downloading and browsing the store, but does not offer remote gameplay support (that's handled by another solution called Remote Play).
Where the PlayStation App shines is as a utilitarian tool for owners of PlayStation consoles, more providing genuine convenience with browsing and downloading from the store to interacting socially. It might not change the gaming experience, but it certainly allows you to stretch that PlayStation ecosystem away from the TV. While the app does have some mild technical inconsistencies and notably lacks cloud gaming functionality, serious PlayStation users will find enough of a value to justify making it their free download.