I have the same issue with this mission. I complete the drive with the tanker and when the video plays and she says 'Hello Mr Whitaker' the screen goes black and the game quits. Its usually the same place it ends however I have noticed that if I jack knife the truck with the tanker attached it also quits out on me during this mission.
Android 4.2.2 on a Motorola XT925 (Cyangenmod 10.1.3)
I have cleared the cache as suggested here
I appreciate running a custom ROM could be part of the issue but I guess the OP with a Galaxy 4 would not be running a custom ROM?