The rules for Travel Pace in the PHB and Basic Rules are good and straightforward, with players able to travel on foot for 8 hours per day without over-exerting themselves, choosing a pace of Slow (2 miles/hour, 18 miles/day; can move stealthily), Normal (3 m/h, 24 m/d) or Fast (4 m/h, 30 m/d, take a penalty to passive Perception).
In this same section, there are two paragraphs about mounts and vehicles; they specify that mounts can gallop at twice the normal fast pace for an hour, and with frequently freshened mounts – typically only available in highly populated areas – a rider can travel fast over long distances. However when it comes to vehicles, only water going vessels or flying mounts get you travelling faster; for commonly available forms of transport:
Characters in wagons, carriages, or other land vehicles choose a pace as normal.
Rules as written it seems there is no benefit to travelling by wagon or carriage, and certainly it takes the same amount of time as walking. In my current game of Dragon of Icespire Peak, the players have spent a bit of time demanding horses or a carriage in order to travel the 65 miles from Phandalin to Butterskull Ranch, believing it would get them there faster (as the quest there seems urgent).
Looking up the travelling rules it really seems there’s no speed benefit to them paying for a ride or hiring horses, except that they might shave an hour off their travel time each day in the latter case. That makes sense for a wagon which is designed only for hauling stuff, not speed, but what about a carriage or a riding horses?
I’d like to know if there are any additional speed benefits (beyond the purely narrative) that I am missing for travel by carriage, or any ways that mounts or vehicles can decrease their travelling time (beyond the double fast pace for one hour). I am aware that a carriage, wagon or even a horse would allow them to carry more weight, but given they have little equipment and don’t need to take anything with them, that’s not of benefit to the party. So I’m primarily interested in if and how they can travel faster, and any non-carrying capacity benefits they might get, for travelling via horse or vehicle.
I’m also interested in any house rules you may have used to give players a speed boost through riding over walking, or why this might be a bad idea.
Best Answer
Horses and carts can carry a lot of stuff
If you only have to travel for an hour, the horse can do so at twice the normal speed. While there is no benefit to speed over a longer distances, horses can carry a lot more stuff than most individual people can.
Any creature can carry 15 times their strength in pounds, and doubled for every size above medium. A draught-horse has 18 strength, so it can carry 540lb (including the rider), and when equipped with a cart it can pull 5x that amount (including the weight of the cart). 2700lb is a lot more weight than most adventurers can carry.
You may think that your 20 strength barbarian would have no worries about carrying all their gear and food, but your 8 strength wizard companion won't be quite as keen once they think about all the extra camping gear and supplies they will need.
That's the main advantage of using horses and carts. You can carry all your food, water, equipment, and other supplies with you in your cart or on your horse.
Using multiple horses to travel faster or further
As you noted, you can travel faster for 1 hour, so if you can frequently swap horses you can travel faster. With mount change you can also travel further. After 8 hours of travel, you risk exhaustion traveling further. You can swap horses and continue traveling without potential exhaustion. Unfortunately, I don't think there are any towns between Phandalin and Butterskull Ranch where the adventurers could swap horses--definitely not enough to swap horses every hour.
Humans are actually really really good at traveling long distances. Horses can sprint very fast, but they aren't faster than people after a couple of dozen miles. Just like in-game, the main advantage is they can carry a lot more stuff than we can, and it's less tiring to sit in a cart or ride a horse than it is to walk.