I've heard people saying the campaign AI is a lot better too. Allied AI factions now do stuff like pronged attacks and such, which I've never seen them do in RTW or MTW2, even on difficult settings.
Yes they do, they even support your own conquests by sending armies to get other provinces/defend yours.
My only question is when you are on the province map is army movement province by province like in Medieval Total War 1 or is it with movement points like in the other game?
No. MTW1 is the last game in the series that did that. Rome onwards is done by 'movement points'.