Local DHT Mode will consume more battery on your device as Jami will need to continuously run in the background to stay connected.
Push notifications through google services or unified push (note unified push is broken atm, see No notifications with nextpush or sunup with fdroid Jami client - #6 by hazematman ) will connect to the DHT network through a proxy so your phone isn’t always connected to the network. This means the app can be closed but you’ll still get notifications about new messages through the proxy. Since the app isn’t continuously running in the background in this mode, battery life should be better as well.