Difference between registerDefaultNetworkCallback and registerNetworkCallback

registerNetworkCallback() was added in API level 21 (Android 5.0, Lollipop). It allows you to listen for changes in networks which satisfy a certain transport type (WiFi, cellular, Bluetooth, ...) and capability (SMS, NOT_METERED, ...).

registerDefaultNetworkCallback() was added in API level 24 (Android 7.0, Nougat) and uses your callback when any network change occurs, regardless of transport type and capability.

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Example for registerNetworkCallback(). At least in my case I do not care about the filters and want my code to run for any network type. This can be achieved with an empty Builder object:

/* Automatically start a download once an internet connection is established */
val cm = getSystemService(Context.CONNECTIVITY_SERVICE) as ConnectivityManager
// could filter using .addCapability(int) or .addTransportType(int) on Builder
val networkChangeFilter = NetworkRequest.Builder().build()
cm.registerNetworkCallback(networkChangeFilter, 
    object : ConnectivityManager.NetworkCallback() {
        override fun onAvailable(network: Network) = downloadStuff()
    }
)

Above code is equivalent to this code with registerDefaultNetworkCallback():

val cm = getSystemService(Context.CONNECTIVITY_SERVICE) as ConnectivityManager
cm.registerDefaultNetworkCallback(
    object : ConnectivityManager.NetworkCallback() {
        override fun onAvailable(network: Network) = downloadStuff()
    }
)

The confusingly named NetworkRequest is actually just a filter for transport type and capability.

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The Default version is less code, but has a lower API level and thus supports fewer phones (74% vs 94%). Since I could not find a backwards-compatible version of this API call in ConnectivityManagerCompat I recommend using the first version if API levels 21-23 matter to you.

As far as I understood, the difference between registerDefaultNetworkCallback and registerNetworkCallback it's only based on customisation.
registerDefaultNetworkCallback works (surprisingly) as a default network listener, while registerNetworkCallback it's more configurable. For example:

    val connectivityManager = getSystemService(Context.CONNECTIVITY_SERVICE) as ConnectivityManager

    val builder = NetworkRequest.Builder()
    builder.addTransportType(NetworkCapabilities.TRANSPORT_CELLULAR)

    val networkRequest = builder.build()
    connectivityManager.registerNetworkCallback(networkRequest, 
    object : ConnectivityManager.NetworkCallback () {
        override fun onAvailable(network: Network?) {
            super.onAvailable(network)
            Log.i("Test", "Network Available")
        }

        override fun onLost(network: Network?) {
            super.onLost(network)
            Log.i("Test", "Connection lost")
        }
    })

Here onAvailable will be called only if the user connects to a cellular network (by connecting to WiFi it won't log anything). The same does onLost when disconnecting from cellular network.

If we do it like this:

    connectivityManager.registerDefaultNetworkCallback(object  : ConnectivityManager.NetworkCallback() {
        override fun onAvailable(network: Network?) {
            super.onAvailable(network)
            Log.i("Test", "Default -> Network Available")
        }

        override fun onLost(network: Network?) {
            super.onLost(network)
            Log.i("Test", "Default -> Connection lost")
        }
    })

Both functions work as default callbacks when the user is connecting (or disconnecting) to/from a network (it can be either WiFi or cellular).

These are just some very basic examples. Of course NetworkRequest can have a lot of configurations by setting its capability or transportType. You can read more about these in the official documentation of NetworkRequest.