Can I copy the app data from one phone to another phone if they both use Android?
You can usually copy the data for user apps without issue, but there is no guarantee.  I wouldn't copy the /data/data/ folder wholesale, since it will include system app data and provides no way to easily manage it.

I recommend Titanium Backup instead.  With root, it allows you to easily backup any subset of apps (such as user apps) that you wish, and then you simply copy over the backup folder to the new phone, install it and select the backup folder, and then restore everything in a single batch operation.  I've used this to migrate apps and data all the way from Android 2.2, to 4.0, to 5.0.