Administration encompasses all the business processes that ensure you get paid publishing royalties for uses already happening - registering songs with global societies, tracking recordings and versions, processing royalty statements, matching ISRCs to ISWCs, managing copyright data, and pursuing unpaid publishing royalties. It's essentially the "back office" of music publishing.
Exploitation means actively creating new opportunities for your music - pitching songs for synchronisation in films and adverts, promoting songs for cover recordings, arranging co-writing sessions, developing creative strategies, securing brand partnerships, and finding new ways to generate publishing income from your catalogue. Full service publishers handle both administration and exploitation, while publishing administrators focus solely on the collection side.