The Storm of Swords expansion changes the style of the game, but not the feel of the game. Your strategy has to shift slightly because having a lot of power tokens handy is even more important to winning this time around. Negotiations take on a new meaning as well, since the expansion gives you leader tokens which can be captured by your enemies. You have to enter hostage negotiations to get your leader back before the Execution card comes up, after which the leader token is unavailable (when my friends and I play, if your leader is executed you also lose his or her House Card -- but that might be a house rule. Can't remember off hand.)
Additionally, there's new Strategy cards which you select at the beginning of each turn. These give you bonuses to pursuing a particular goal, such as conquering territory, holding territory, manuevering troops, securing and rescuing hostages, possessing key sites, forging alliances, and supporting allies -- though the Support Allies card is only available in the full version of the game. Forging Alliances is an interesting card to play; it lets you gain some extra abilities or units based on ally cards from either House Tyrell (who are unavailable to players in this version of the game), House Arryn, House Frey, and mercenaries. Ally bonuses include extra troops, extra supplylines, the ability to cross otherwise uncrossable borders (Weather comes into play with the new rules, so that rivers crosses are occasionally flooded and inaccessible), gaining additional leader tokens, and a variety of others. You can solidfy your alliance by possessing three cards from one of the four Ally decks; but you have to watch out for your opponents buying off your allies and causing them to betray you.
I think that pretty much covers all the changes with the new rules. The reason it seems to move quicker is that there are more ways to win than just hold key territories. You have to earn a total of 8 Claim Westeros Points, which can be picked up by holding major cities, by following certain strategies, by winning over allies, and by tossing lots of power tokens around and just buying your way to the top.
Clayton.