Both are agreements
An embargo asks several outlets to hold an item until a stated time. An exclusive offers one outlet the item ahead of everyone else. Neither is enforceable; both are conventions that hold because they are useful to both sides, and both collapse the moment one side treats them as an instruction.
What an embargo is for
Time. A reporter given material three days ahead can read the underlying documents, seek an independent view, and file something considered rather than something typed at speed. The organisation gets coverage that is better informed, and everyone publishes at once, which suits announcements where simultaneity genuinely matters.
What breaks an embargo
Sending it to people who never agreed to it. An embargo offered in advance and accepted is an agreement; an embargo asserted at the foot of an unsolicited message is a request that carries no weight, and treating a publication as bound by one it never accepted is the fastest way to lose the arrangement permanently.
Setting a time that suits only the sender also breaks it in practice, as does breaking it yourself: an organisation that publishes its own announcement before the stated hour has released everyone else from the agreement.
What an exclusive is for
Depth. One outlet given sole access will usually invest more: more space, an interview, a photographer, a piece that runs longer than a news item. The price is that every other outlet now has a story someone else broke, and most will not cover it at all.
Choosing between them
The question is whether the aim is that many people hear something, or that a particular readership understands something properly. Announcements with wide public consequence generally suit an embargo. A complicated development that needs explaining usually suits an exclusive, provided the outlet chosen actually reaches the people concerned.
Being honest about which is which
Offering the same item to three outlets as an exclusive is the one failure with no way back from it, and it is always discovered. So is describing something as an exclusive when it has already been sent under embargo. Both arrangements rest entirely on the sender’s word being reliable in small matters.