The APK from Aurora Store is the Google Play version. The only difference from the Google Play version and the one from the website is that the latter has a built-in auto-updater.
The APK from the website is usually updated last, hence the slight delay.
Are there any issues/concerns in general for apps directly downloaded via apk vs using aurora store? I know telegram for instance has censorship on the aurora/google version that isn't present on the direct apk.
There are no particular concerns for app updates. For first installs, an app store may offer more security guarantees than just relying on the web PKI (regular HTTPS) to ensure the authenticity of the downloaded APK.
As for app functionality differences based on source, there is no general rule since this depends solely on the app and should be documented by the developer. Apps outside Play Store may or may not bundle Play libraries, there's no particular rule. Signal from Aurora/Play already has a fallback for push notifications (its own websocket much like Google FCM) in case Play services isn't available. Element from Play Store does not, while Element from F-Droid has an alternative (by periodically fetching content).