Difference between Google API intel x86 atom and Google Play intel x86 atom sytem image

The play image has the Google Play store already installed and you can install apps, but there is no root access as it is a production build and not a debug build.

The main difference between Google API system image and Google Play system image basically lays on the capabilities of each one to access the Google Play Store directly or not. Here is for more explanation regarding that matter.

Hope, could be a helped for the other.

"Google APIs" images include the Play services that many apps rely on but no Play Store app

"Google Play" images include both the Play Store app and the Play services

For Google's description, see Create and manage virtual devices | Android Developers

What all answers are missing: If you use the billing library from Google, you need the "Google Play image" for testing. This has nothing to do with installing apps, but you need the Play Store installed to test your "in-app" purchases and subscription products. Otherwise you will get this error from BillingClient:

Billing service unavailable on device

Based on Documentation:

A system image labeled with Google APIs includes access to Google Play services. A system image labeled with the Google Play logo in the Play Store column includes the Google Play Store app and access to Google Play services, including a Google Play tab in the Extended controls dialog that provides a convenient button for updating Google Play services on the device.

To ensure app security and a consistent experience with physical devices, system images with the Google Play Store included are signed with a release key, which means that you cannot get elevated privileges (root) with these images. If you require elevated privileges (root) to aid with your app troubleshooting, you can use the Android Open Source Project (AOSP) system images that do not include Google apps or services.

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