App developers have the ability to do a staged roll-out of updates to their apps through Google Play Store. This means that a small percentage of users (chosen randomly) receives the update first. The developer can gauge the feedback, and then proceed to a wide roll-out until they reach 100% of their install base.
It's possible that the latest version of this particular app was recently released in such a staged roll-out. One of your phones was randomly selected to be in the initial stage, while the other was not. There is nothing for you to do, other than wait for the roll-out to complete.
As comments suggest, you can also attempt to sync the app manually by retrieving it from one phone and installing it on the other. This requires installation of USB drivers for your phone on your computer, and a copy of Android SDK. If you are rooted, you can get the APK of the app using
adb pull /data/app/<app_package_name>.apk c:\ command, and then connect the other phone and do
adb install -r c:\<app_package_name>.apk.
On unrooted devices you may want to look into
adb backup utility which allows you to generate an
.ab file which contains the APK of the app, along with its data. For example, the command
adb backup -f c:\Skype.ab -apk com.skype.raider creates "Skype.ab" file on my C: drive that contains the Skype app and its data. I can then use
adb restore c:\Skype.ab command to restore the app with the data. Optionally, you can clear the data for the app from your other phone in Settings -> Apps.
The difficulty is locating the package name of the app you want to back up. Although package names are usually descriptive enough, sometimes they can be quite cryptic. Fortunately, Play Store itself has the package name in the URL. In your example, the app Out of Milk is located at the following URL: https://play.google.com/store/apps/details?id=com.capigami.outofmilk. The portion of this URL after "id=" is the package name. In this case it's com.capigami.outofmilk. So your backup command becomes
adb backup -f c:\OutOfMilk.ab -apk com.capigami.outofmilk.