How do the expansions for Carcassonne change the game?
Abbey &amp; Mayor

Introduces 6 "joker" abbey tiles. Six Mayor meeples, one for each player. Twelve new landscape tiles. Six wagon pieces, one for each player. And six barns, one for each player.

Wagons allow you to move from a road to an open city tile if you complete the road. This allows you to claim something new without using an extra turn.

The Mayor meeple can only be used on a city tile, and counts for an extra follower for every shield found on a city tile.

The six Abbey tiles allow you to fill a gap between a cross of tiles. It has to be surrounded from top, bottom, left, and right (diagonal is not necessary).

The barns allow you claim a farm on any corner where 4 green fields connect. Any farmers present on the farms at this point will get returned to their owners, and normal scoring rules will apply. The barn will remain, however, and score 4 points for each city at game end.

I think Abbey &amp; Mayor has the most impact of all the expansion packs. 

One of challenges of Carcassone was farming. Farming was a critical component to winning. But committing too many farmers, and getting into a farm-war early on, was quite expensive. You would lose many of your meeples until game end, which prevents you from using them at a critical junction later in the game. Striking the right balance between farming and other objectives was critical. 

The barn invalidates this and allows you to be more reckless in the beginning to get a "point rush" until your farmers are eventually returned to you, freeing them up for other uses. Personally I'm not a fan of this feature; it places far too much emphasis on luck, and being the first to connect a barn is the most advantageous. This alters the beginning of the game where it's a race to farm and place a barn.

The Mayor meeple changes game fundamentals too. For every shield the Mayor meeple counts as an extra follower. This does make for interesting cut-throat gameplay. The game becomes far more aggressive where everyone is actively trying to steal others' cities.

This is also something I'm not pleased with. I feel, over all, the Abbey &amp; Mayor shifts the balance of the game to a far more aggressive posture, and this could be annoying to some as they will rarely be left alone and must respond to aggression on every turn instead of trying to play their game.

Finally, the "joker" card, or Abbey tile, I feel is a cheap "get out of jail" card for the player, and invalidates some of your strategy if you like blocking your opponents while you work on your own objectives.

I am extremely competitive but I feel the game already has a fantastic balance with other expansions (Inns &amp; Cathedrals, Builders &amp; Traders) and this expansion just tips it a bit too far. My friends like this and as such I play this expansion very often. 

Inns &amp; Cathedrals

Introduces 18 new landscape tiles, and six large followers. The new tiles include two special cathedral tiles.

The landscape tiles now include Inns, which, if the road gets completed, let you score two points for each tile instead of one. This is a nice feature as it adds more value to road tiles, and shifts some of the emphasis from city building to road building. These tiles come with the added risk of, if they're not completed, not receiving any points for them. I think this is a nice feature in terms of risk/reward. It's very cool to watch others trying to join your long road that has an Inn for a share of the spoils!

Cathedrals applies the same concept to city building. The reward in this case is three points for each completed city tile, and no points for an incomplete city. This is fascinating because you can use cathedral tiles offensively to prevent opponents from earning points on their cities.

The large follower, which counts for two followers, is also intriguing since it presents you with a chance to steal your opponents points by taking over their cities, farms, or roads. You now have an incentive to be aggressive, whereas in the original Carcassone such insidious plotting required more careful planning and an extra turn to place two followers.

If you include Abbey &amp; Mayor such strategies are somewhat mitigated since the Mayor will likely trump a large follower (the Mayor can only be used on a city tile). The Abbey, or "joker" card, will go some way to allow a player to complete their feature even if you strived to make this task difficult.

Traders &amp; Builders

Introduces 24 more landscape tiles. Twenty trade good tokens (9 barrels, 6 grain, and 5 cloth). Six pig meeples, and six builder meeples.

The pig meeple is a great addition. It allows your to earn 1 additional point for every city connected to a farm. You must already have a farmer on the farm in order to place the pig.

The builder meeple is a fantastic addition and changes the gameplay substantially. You can add your builder to any city or road with a follower. On your next turn, if you can add to your city or road, you get an extra turn immediately for a maximum of two turns.

This is great because it adds an element of risk that plays into the largest part of the expansion: The trade goods. Each of the new added tiles has an icon denoting a trade good. The player who closes a city, irrespective of the owner, gets to keep whatever trade good tokens are denoted on the city tiles. If you close a city with two tiles that have barrels on them, for example, you get two barrel tokens. This is exciting because having a builder on a city gives you an extra turn that increases your chances of completing your city and gaining the tokens.

Whoever has the most of each token by game end earns 10 points for each type. If you have the most barrel and cloth tokens you gain an additional 20 points to your total.

This adds an incentive to close your opponents cities if you deem it beneficial to you, or damaging to your most threatening opponents. Closing a city with trade goods could deny your opponents points, or/and help you earn more points, or match their number of trade good tokens, thus invalidating their advantage. A lot of thought can be applied to how you manage this feature of the game.

I haven't played any of the other expansions, but I know of Tower, River II, Count, King &amp; Cult, Princess &amp; Dragon, Wheel of Fortune. I will give Tower a try this week and update my answer.

That said, of all of those I think the Count sounds most fascinating and will likely change gameplay drastically. It includes a separate set of pre-numbered tiles that must be assembled to create a castle. Your followers presence in the castle now and other events that take place on this castle now influence gameplay and scoring.