In the stone age we would have had less food.
From the bronze-age onwards things are fairly different - the great horse cultures of southwest asia never arise from the Scythians onwards. The Hun and Mongol hordes have to sweep over the west on foot ( or possibly on camels ). The wild vitality of these people does not drive innovation across Europe time and again. The Persian messenger services can't move as fast and so, logistically speaking, empires are likely to be smaller as communications are slower. The world becomes a bigger place.
Mythology has neither unicorn nor centaur.
There is a chance that with only camels or elephants as riding animals, north-western Europe does not achieve the kind of prominence it did in the age of knights - the medieval cavalry never takes the field and strategies that are designed to combat it ( schiltrons etc ) also do not arise. Harold holds England against the Norman invasion. Heavy plate armour is less practical so lighter chainmail is standard. Possibly camel-mounted peoples from the south would be able to sweep in with rapid attacks and seize large areas of Europe and Asia with very little strategy available to combat them at first. This would certainly change the shape and history of the west significantly.
Without the superiority of the knights the crusades might have been harder to achieve and the consequent movement of the plague across Europe may have been slowed, meaning that there would be fewer smaller outbreaks rather than the huge ones that wiped out vast swathes of the population in the middle ages.
The horse and carriage never arises which prevents another form of rapid travel that carried people over longer distances. The difficulty of landward journeys being so much higher would probably favour seafaring as the next most effective way of travelling. People would travel by sea to get between cities in the same country because it was simply faster. This might lead to canals being built and favoured rather than roads and a more water-oriented civilisation.
In farming, oxen would replace horses and donkeys for heavy field work - they are slower and generally less precise, but it's possible that with selective breeding this could change.
Given the influence of equines on the last few thousand years here in the western world, I think a history without them would be more different than most people who haven't studied the field realise.