They are two different things actually. Back4app is a managed service. They both manage database and parse server. If you ever decide to host your own server that means you have to manage your own server. Both database and parse server. You have to scale your app yourself. While there are guys in Back4app that get paid to do this for you.
One more tip: AWS is very complex environment. Be careful when using it. There are lots of pricing criteria. If you don’t have any experience with aws or you don’t know what you do, you may face some suprise bills.
Also Back4app owner is a member of this community. And he is very helpful. I believe he can give you the information you want. @davimacedo