ADB is part of Android SDK. It allows you to debug a device connected with a USB cable. The problem with ADB is that you need to install a driver specific to your device. Some devices do not support ADB at all.
B4A-Bridge doesn't require any driver. You install a service that runs on the device and communicates with the IDE over the local network.
When you install an application with B4A-Bridge you need to approve the installation. This is a security limit of Android.
If your device supports USB debugging then you do not need to use B4A-Bridge (except of special cases like USB host development).