City folk tend to see horses and cows as essentially quite similar animals. They're similar in size and they both eat grass.
However, they are quite different in many ways.
Cows (and sheep and goats) are ruminants. They have a four part stomach and they eat each item of food twice. It's partially digested, regurgitated, chewed again and swallowed into a different part of the stomach - a process called "chewing cud". (Rabbits, incidentally, perform the same process by a slightly different mechanism - coprophagy - they produce two kinds of dung, one of which goes through again).
Horses are hind gut fermenters. They have a relatively small stomach and a massive gut, and they handle grass by digesting it for a long time in a large gut that ferments it extensively. (This, by the way, means it is very, very hard to intoxicate a horse, even a small one.)
Cows have cloven or divided hooves. This means they stand on two toes of the original five.
Horses stand on one toe - a single hoof.
These differences are key to our different uses of these animals.
Cows have been used as work animals in the past, because it is easier to create a pulling harness for oxen than for horses. Oxen can pull using a yoke, which is similar to the yokes used to increase the lifting capacity of men. Horses, because of their longer and more flexible necks, can't. But as soon as people come up with a good way to use horses, they mostly stop using oxen.
An ox team pulling using a yoke. Source: Allen Drebert via Wikimedia Commons.
It comes back to that chewing the cud thing. A cow's digestive system has to stay constantly active. She has to have enough fiber in the rumen (first stomach) for it to stay active and healthy. This means that a cow has to spend at least eight hours a day, every day, chewing cud. If you feed cows high energy feed to reduce the amount of time they spend eating, you can damage their rumen - a balance dairy farmers battle with.
Horses, in the natural state, spend about 20 hours a day grazing. However, you can give a horse high energy feed - grain, carrots, apples - a couple of times a day and reduce that grazing time down to a few hours a day...meaning you can work a horse for 8-10 hours a day and have it stay healthy. It's better for them to spend all day grazing, but they aren't going to get sick off of spending less time eating and more time working alone.
So, you can only realistically work a team of oxen for four hours a day - and you can work a team of horses for eight. Which gets your fields plowed faster?
Hence, we don't ride cows.
Because there are exceptions to everything - a rodeo cowboy trying to ride a bull. Source: Cszmurlo via Wikimedia Commons.
There are also a few people riding and even jumping cows out there. I wasn't able to find a legal-for-use picture, sadly.