The agriculture in the Americas is quite advanced, even on the Great Plains (look up "Corn Culture" and it's amazingly different from our 19th century view of the plains tribes.) Horses meant hunting got quite a bit easier and did moving to more comfortable or less depleted locations over a far greater range...the nomadism is about capacity and dogs just couldn't pull much of a travois load while a small extended family of 8-15 people was what hunting and gathering would support on foot (vs. much more populous permanent villages in earth-lodges with large fields of diverse crops at higher yields than Europeans were getting.)
If horses didn't die out and instead were contemporaneous throughout the Indians' presence on the continent, they're at least a common meat source and domestication into herds wasn't nearly as active of a practice in the Americas as it was in Eurasia and Africa (cattle, sheep, goats.) Making that mental journey from easy hunting prey to faithful personal transportation is a long one even in Eurasia (look at breeding for size/characteristics, saddles/bridles/stirrups/wagon harness, veterinary care, horsehoeing, horse-based weapons/tactics, wagon/cart/chariot development.) But the tribes would have 10,000+ years to figure it out (and probably domesticate American wild camels too which died out when the horse did I think.)
Trade networks for greater volume and real roads developing in North America like the Mayans built, mechanizing some farming with horse-drawn equipment, horse power for windlasses, pumps, etc., faster and further migrations from drought-stricken regions, more contact with other tribes and other bands within the same tribe would be possible impacts.
So the Spanish would have had a vastly tougher time in the American Southwest from mounted Indian warriors (but most of Coronado's opponents were succumbing to the Spanish troops' communicable diseases rather than knight's lances or crossbows.) So much of the earlier fighting would be in places along coast-lines, heavily forested where cavalry, let alone random clusters of mounted warriors are much tougher to use well.
The Indians would have certainly had a happier time of it, nobody romanticizes dirt farming or trying to lure hundreds of bison off of a cliff for the upcoming meat supply.