Something occuring exclusively is only happening or available in one special circumstance, like a song that’s exclusively for sale on one web page, but not anywhere else.
Something exclusive is only available in one way, like a newspaper with an exclusive interview no one else was able to get. Things happening exclusively are also available in a select way—not for everyone. If a boxing match is shown exclusively on pay-per-view, it’s shown only on pay-per-view. If I work exclusively for one company, I don’t work for any other companies. If a singer sings exclusively at one club, that’s the only place to hear her.