Where do you store your videos? Are they on the mini hard drive? Are they on an external? If so, how is the external connected? Are they on a network drive? If so, where is that drive and how is it connected?
Then answer the other questions:
Are you using Plex, VLC, or something else?
Are you using the same movies you had encoded for the ATV or did you encode your movies again once you moved from the ATV to the Mini?