The Tracking API pulls shipment status on demand or by webhook. A request against a tracking number returns the carrier history AfterShip has aggregated, and a webhook can push a status change the moment it happens instead of making your system poll for it.
The Shipping API, still called Postmen in places, handles rates and labels. It quotes a shipment against whichever carriers you've connected, returns an option by whatever rule you set, cheapest, fastest, preferred carrier, and generates the label, the same job ShipEngine or EasyPost do for other businesses.
Authentication is a single API key from the AfterShip dashboard. There's no OAuth flow to set up before the first real request goes out, which is part of why the gap usually isn't getting access. It's deciding what the rest of the business should actually do with what comes back.