Event Promoter automatically imports your event RSVPs and people who have purchased tickets for events. It is hooked up to the events calendar plugin on the website which you use to publish information about new events.
Mailchimp contains your active member directory, which is updated automatically with member info every time someone joins, as well as your inactive member directory.
There's some overlap between the two but these programs don't communicate unless you manually import people from Promoter to Mailchimp following an event using these instructions.
Promoter is for getting in touch with event attendees (e.g. to send them details about the event, send them a notice if the event is cancelled or changes, follow up with them afterwards, etc.)
Mailchimp is for contacts like your active membership, expired members, and non-member database.