Lets just cut to the chase: what if Europe/Siberia had its own native elephant-like species that was available to be tamed and used for all the uses elephants were/are used for in those parts of the world in which they live. Mostly prestigious war animals, as a spectacle, or a few isolated tasks requiring loads of brute force. We'll assume that the species in question has evolved to keep pace with the temperatures, so likely less wooly than the classic mammoth image in our head.
I imagine the biggest change would be the use of war mammoths in fashion similar to war elephants, so expect Celtic and Germanic peoples to usually have a few mammoths among their larger armies. This might make them more durable against Mediterranean civilizations, or it might make them more appealing conquests - they'd likely be somewhat wealthier with an additional resources like mammoths, who also are likely to deforest the land to a marginal degree, which also makes them a bit more easy to conquer.
I imagine the biggest change would be the use of war mammoths in fashion similar to war elephants, so expect Celtic and Germanic peoples to usually have a few mammoths among their larger armies. This might make them more durable against Mediterranean civilizations, or it might make them more appealing conquests - they'd likely be somewhat wealthier with an additional resources like mammoths, who also are likely to deforest the land to a marginal degree, which also makes them a bit more easy to conquer.