As you've said, the GeoIP2 Java API is licensed under the Apache 2.0 license. MaxMind has clearly granted you a license to distribute and modify the software under those terms. The terms of the Apache license are quite permissive and primarily only require attribution.
The statement by MaxMind that their "GeoLite2 Developer's Package" includes a license to include their Java API in a product does not in any diminish the fact that they have already given you a license to do exactly that, via the Apache license.
Why would they include this? For one, it lets them add another marketing item to make the developer's package sound more attractive. For another, the license grant included with the paid package does not carry attribution requirements or the requirement to include a copy of the Apache license. Also, prior to Jun 5, 2013, the API was licensed under the GNU GPL, which is not a permissive license. (I don't know if they even offered a paid alternative in 2013, though.)
It is interesting to note, though, that the paid package's license grant includes the requirement to assign copyright for your modifications to MaxMind:
You have the right to modify the Named MaxMind Products and redistribute the software with the changes; however, any such changes must be relayed back to MaxMind, and copyright in them assigned to MaxMind.
Presumably, you could keep copyright in your changes by electing to modify the API under the terms of the Apache license instead, but you would then need to include attribution and the license text in your application. The other components (database, etc.) appear to only be available under the paid option, however, so you have no choice in terms for those works.