The Nitrogen in horse manure is more readily available but at 0.6 (cow) and 0.7 (horse) there is not a significant difference between the two. Both work very well in compost piles and niether is better.

Ruminants are better at nutrient extraction than horses, and a usual benefit of that is destruction of seed.
But the quality of manure in any case will be determined by what the inputs are.
If the horse(s) are eating differently than your cattle, the manure will give your compost a potentially broader range of desired substances.

Thanks! This is what I needed to know.

I guess that's not all. My friend is just certain that horse manure is "hotter" than cow manure. She thought that meant it had more nitrogen, but now she thinks maybe it means it has more ammonia. I read a little and it seems like the ammonia affects the pH and makes it more acidic?

I think I agree about horse manure being 'hotter' than cow. Makes sense, since I assume a cow's four stomachs extract more nitrogen than a horses' one.
I don't use manure after finding it was pushing my high phosphorus levels through the roof.
I've also imported some unwanted plants to my place in horse manure.

"She thought that meant it had more nitrogen, but now she thinks maybe it means it has more ammonia."

"Ammonia" is the form the nitrogen has taken. To my knowledge ammonia buffers acidity and any pH effects are transient.

The N content will vary depending on what the animal has been eating. For instance a cow from a poor pasture will not have much wasted N in the manure. But manure taken from a cow lot may have a lot more. This manure will not only have more wasted N but also have a lot of urine mixed in. It would hold true for horses also.
Having said this, if both are composted with browns then the N will be mostly be used up in the composting process.
It is always suggested that all manures be well composted before use.
On another note it was said in another post the seed will be killed off in the rumin system. I add different seeds to cattle supplement to spread it over a pasture. In nature this is how a lot of seed is spread.

I have used cow & horse, both are good.
I have used chicken,rabbit & hog also.

Horse manure tends to have a more readily available Nitrogen then does cow manure so it is "hotter". The amounts in each, from years of analysis, are about the same at 0.6 and 0.7 percent. I have not noticed much, if any, difference in compost made with cattle manure or compost made with horse manure in all the years I have been composting, or in compost made with rabbit, poultry, Llama, Alpaca manure or blood meal or synthetic fertilizers.
As long as the mix is roughly 3 parts vegetative waste to 1 part manure with just enough moisture (too much water replaces the air also needed) the bacteria will get to work and digest that material. They really do not care about the source of Nitrogen.

Both work, but I found there to be a whole lot more seeds surviving in horse manure than with cow manure. Whatever the horses have been eating will start sprouting everywhere, but if you are aware of it and just work to get it well composted then no problem.