Overheard yesterday:
If I can't use Duo Push, then I may as well use Google Authenticator.
Duo Mobile is a native mobile app. Duo Push is a capability of Duo Mobile on certain devices (iOS, Android, etc.). Duo Mobile protects two secrets: a symmetric key and a 2048-bit public/private key pair. The latter is used to secure the active, online, Duo Push authentication protocol.
In addition to Duo Push, the Duo Mobile app supports standard OATH TOTP, which requires a symmetric key. One copy of the symmetric key is protected by the Duo Mobile app while another copy is protected by the Duo cloud service. The two work together to provide a standard, offline, one-time password authentication protocol.
Google Authenticator supports OATH TOTP as well but the similarity with Duo Mobile ends there. Google offers no service analogous to the Duo cloud service, nor is there a server-side app you can run on-premise (but check out an open-source OATH TOTP server developed by Chris Hyzer at Penn). Google Authenticator does come with a PAM module but it is not enterprise grade by any means.
Bottom line: If you want to install an app on your mobile device that supports OATH TOTP, choose Duo Mobile. It's free, it works with all OATH TOTP-compatible servers (Google, Dropbox, AWS, etc.), it is well supported, and it does so much more (such as Duo Push, if and when you need it).