There are differences in function, not just in purpose.
Entering/exiting
- Kid's Corner is more convenient to access. You enter Kid's Corner by swiping left from the lock screen. To switch back, don't swipe left at the lock screen. Apps Corner is accessed by pressing the Launch button in Settings > Apps Corner. You return to the regular profile by holding the power button and swiping right.
- Apps Corner is slower to load. The delay between leaving Apps Corner and loading the Start screen is several seconds, while it's around two seconds with Kid's Corner (on the Lumia 521).
- Apps Corner doesn't have a lock screen when you press the sleep button.
Adding and removing apps:
- Apps Corner can have more apps enabled. With Apps Corner, you can't add Store, Settings*, or Data Sense. With Kid's Corner, on top of those three apps, you can't add several more first-party apps like Files, Calendar, Email, Office, OneNote, Internet Explorer, and Cortana/Search. [*Settings is available through the action center, but with vastly fewer options.]
- Adding apps to Kid's Corner is more convenient. You add apps to Kid's Corner by long-pressing the app in the app list on Start, though you have to go through Settings to remove apps. With Apps Corner, the only way to add (or remove) apps is through Settings.
Multitasking, actions+notifications
- Kid's Corner can't multitask. You can't access the multitasking screen (long-pressing the Back button). When you switch to another app, the first app is closed. (Not sure about whether background tasks work.) Apps Corner multitasks normally.
- Apps Corner can have the action center enabled or disabled. Kid's Corner can't have the action center.
Start screen and customization:
- Kid's Corner can have a different theme (white/black background) and accent color from the regular mode. Apps Corner can't have an independent theme or color, but I think it can have a background picture?
- Kid's Corner preserves rearranging and resizing the tiles. While for Apps Corner, it's not preserved when you leave Apps Corner
- You can export your settings for Apps Corner to an SD card. Export is Settings > Apps Corner > Advanced > Export to SD card. Import is Settings > Apps Corner > Import from SD card (that button appears only if the SD card has Apps Corner data).