So presumably the AI civ's aren't getting the same influx of events that players do, so there's a lack of parity to some extent. AI civ's get to make demands and offer deals that we the players can't initiate, at least not with their ease. But I guess it washes out if we get positive events that they don't. But I don't know that AI civ's get to have council members of their own, or perform actions like influencing or tutoring or gifting luxuries.
An ambassador can initiate a trade mission at the cost of civics, at the cost of some influence. It is usually a bad deal that frequently is not worth taking, which makes it not worth the costs of the civics in the first place, but your ambassador does get some XP they wouldn't otherwise.
Likewise, you can ask an AI civ to declare war on another civ, BUT unlike such demands they can place on you, you can only do so with civ's with you are currently friendly.