How can I sync photos between iPhone 5s and iPad mini without using a laptop? I can do it using a laptop. But is there a way I can do it without passing them through a laptop each time?
In my understanding, I suppose that Photo Stream will do the trick.

Go to Settings > iCloud > Photos > My Photo Sream, and turn it on.

Enable iCloud on both devices using the same Apple ID, and you can see the latest photos in Photos.

Apple - iCloud - Photos