Comparing the Android we have today to what was available several years back is stark not just in terms of UI. Google has addressed many pain points in the realm of usability and features over time. Many of the things we used to need root access to get done are now possible on completely stock devices, even on the stripped down Nexus variant of Android. One notable exception is the state of application backup on Android. It's an absolute mess, and Google has tried to fix it with little success. Let's go over your options and find out where things stand.
What is app data?
When people talk about app data, they are usually referring to the content stored under each application or game's folder in the system directory of Android. You can see how much data an app has accumulated by going into the application settings. Android gives you the option to delete this data, but that's all. If you do so, it reminds you that you're going to lose all your settings, accounts, and so on. That's what we're talking about -- your stuff.
For an app, this directory might contain your account information for an app that needs you to log in. It also contains any data you've input into the app since you started using it. For example, a fitness tracker app will have all your workout records and history. If you delete the app or clear the data, that's all gone. The developer needs to specifically make allowances to back that data up in such instances (more on the alter). For games, the app data folder contains save games and settings. Again, if you delete the data or uninstall the game, your progress is gone with it.
So why can't you simply copy the data from these directories and save it somewhere? App data is all in the system partition, meaning you need to have root access to do anything with it. That might seem like a kick in the pants, but it's a common security measure. You don't want one app being able to just snoop around in the data of another app. The only way to back up and restore app data is through rooting or a system component. Google has thus far really dropped the ball on the latter.