Tactics Advanced is different compared to the PS1 game. The story's not the same at all, with none of the political intrigue and it's a more lighthearted in tone in comparison, and I think the battle system is a little more simplified, if I remember correctly you can cast spells instantly in Advance but in Tactics you have to wait a number of turns. The biggest change between the two is Advance's Law system, where each battle is governed by a set of rules like you can't use this type of weapon, or these elemental spells where if you break them during battle your character can be imprisoned. But each battle has settings that Judges like to see in battle, and if you fulfill those conditions then you get Judge Points which can make combos with allies or activate summons.
Still a great game, but for me, in terms of the better story and the depth in the fighting, I prefer the original. I just love the world of Ivalice. If you want to play that one Shi then you can either play the PS1 version, or ideally the upgraded edition they released for the PSP known as FFTactics : The War Of The Lions, which adds multiplayer, animated cutscenes, voice over work and I think new job classes and new special characters to recruit like Baltheir. Can't really go wrong with either one, both has some really addictive gameplay.
I really want to go back and play it now! /dusts off his copy