We don't really know.
In the Store, they describe the Horse as suitable for 'riding short distances' and the Thoroughbred for 'long distance travel.'
That said, given the existence of the Ursaphant and the Trison...there probably aren't heavy draft horses and heavy warhorses in Elyria...because they would be heavily outclassed by those two animals.
So, by my best guess...a Horse is your 'jack of all trades' animal, while a Thoroughbred is more specialized in long distance travel (lower carry capacity, much higher endurance)
I did write up a 'proposal post' reflection how I thought the different mounts should interact here, if you're interested in having a read.