Many new mechanics have been added to them like the ability to buy and sell them on market, summon them to your location when needed, and upgrade their stats for running, jumping, and overall health.
Horses are a very personal possession, but be careful. If a horse dies, it will be dead for a total of 7 days (real time). Afterwards, it will respawn for use again.
Talk to the Stablemaster who will sell you a horse, donkey, or mule of your choosing along with other saddles and tack.
Once you have bought and used a horse, donkey, or mule voucher, you can access you stable by doing /horse
A GUI will open where you can see your owned mounts. To spawn one, left-click on it, or to access it's settings for upgrading or renaming, right-click on it.
To upgrade your horse's stats, you have to use it. 1 XP = 1 Block Travelled. Upgrade them by clicking on the 'Upgrade Horse' button when your horse is at max XP. Each upgrade costs some gold, so be sure to pad your pockets on your journey.
To see mounts listed in the market place, do /horse market
To list your own mount, manage it's settings in the GUI menu and click “sell horse”.
Happy riding!