As noted, anatomically modern humans evolved before the last ice age. The evolution of behavioral modernity took place about 50,000 years ago, and is still not fully understood. Perhaps the stress of living in an ice age environment promoted the traits that led to behaviourally modern humans.
The other possibility is that the ice age would lead to physical evolution, as small bands of proto humans are isolated by ranges of glaciers and live in different environmental ranges. Humans trapped in Europe might eventually evolve in ways similar to the Neanderthal people, who were physically changed in ways that favoured living in a cold climate (large noses to pre warm the air they breathed, shorter bodies and limbs to reduce surface area, etc.). Modern Inuit people have similar adaptations, and if they had been isolated more completely and for longer periods of time, they would probably have developed further adaptations to the cold.
Other groups of protohumans might have become adapted to foraging across the great steppes and savannahs of Asia and Africa, since the cold dry global climate would have limited the growth of forests. These humans would be much taller and long limbed, to allow them to cover long distances and run down prey. Once again, we see hints in some groups like the Maassai, who are generally much taller than similar populations.
How extreme these adaptations become wold depend a lot on factors like how hard and deep the ice age is, and more important, how long the various populations are isolated from each other. If we wait long enough, then these peoples will become separate species (although this might take tens of thousands of years).
One other factor to take into account is you suggest this takes place as the protohumans become technologically proficient. Even with Paleolithic technology, humans were able to adapt and overcome pretty extreme environments, including the arctic, and literally walked around the world, crossing oceans and seas along the way. This sort of adaptability helped humans survive and become the dominant species they are today, but also makes having isolated populations able to evolve in peace very unlikely. Homo floresiensis seems to have evolved into their final form after being isolated in their island archipelago anywhere from 100,000 to 1,000,000 years ago (there is a great deal of uncertainty about when they arrived and indeed what their actual evolutionary background is), but were apparently never in contact with our own Ancestors between their arrival and extinction 12,000 years ago. Other human species like the Neanderthals and Denisovans were indeed in contact with our Ancestors, and most peoples in the modern human family were in limited contact with each other through trade and exploration even in ancient times, so there was enough interbreeding and movement between populations to prevent humanity from evolving into different species.