See the little box in the the lower left with the check mark in it? Toggle off the check mark.
The new Start dialog (which replaces the older Welcome dialog) is new in the After Effects 13.6 update. If you have not updated After Effects CC 2015 on both machines, you'll see this difference.
You can control whether or not the new Start screen is displayed when you start After Effects via a new option in Preferences > General, Show Start Screen at Startup. (This is enabled by default.)
(Another reason that you may see the old Welcome dialog is if you are using After Effects in certain languages. I don't remember the list, but because this is a new feature it has not been localized to all supported languages yet. But it looks like you're using English, so I don't think that's the case for you.)
Many thanks, Tim. I had thought I was up to date on my laptop but just checked and I'm not on 13.6, so that's the answer. Thanks as well, Dave, for responding.