After a short soak test, Motorola and AT&T are finally rolling out the Android 5.1 update to the 2013 Moto X. While other 2013 Moto X's received it quite some time ago. It's okay to be last, as long as there are no bugs. The update is rolling out to everyone now, so head on into Settings > About Phone > System Updates > Check for Updates. Remember that AT&T only allows you to check for updates once every 24 hours, so you won't be able to check again until tomorrow.
Some of the changes in this update to Android 5.1 include Material Design, as well as Smart Lock. There's also the revamped notification drawer, which should make things easier on the 2013 Moto X. With the change to Quick Settings, it's just an extra swipe down now to get to those Quick Settings, compared to the two finger swipe or tapping the icon in the corner, previously. There's also the interruptions in place of a silent mode. Which is nice, but most people would prefer silent mode.
With Lollipop, you get Ambient Mode, which is almost redundant if you use Moto Display. However, you can quickly run into the Settings app and turn off either Moto Display or Ambient Mode depending on which one you like better. Which is one of the perks of using Android. Remember that now you're on Lollipop so you'll be using the Android Runtime instead of dalvik. Which means it should run faster, however, it will take a bit longer to start up. You can check out the full lengthy changelog in the source links below.
If you do get the update, it's a good idea to backup your data and do a factory reset after you've updated. Especially coming from KitKat to Lollipop. As there are often times some issues that arise from migrating data from KitKat to Lollipop, and a factory reset will easily take care of that. So if you are having issues after the update, that's something you should try out.