How is climate change today different than the climate change that ended the ice age(s)?
Difference #1: Speed. The warming at the end of the ice ages went much, much slower than what we are currently experiencing.
Estimates of the amount of warming since the beginning of the industrial revolution are in the order or 1 degree Celsius* - that's in 100-150 years time.
* Page 7

When the Earth moved out of ice ages over the past million years, the global temperature rose a total of 4 to 7 degrees Celsius over about 5,000 years.

(NASA Source: How is Today’s Warming Different from the Past?)
The article Earth is warming 50x faster than when it comes out of an ice age sums it up nicely:

What humans are in the process of doing to the climate makes the transition out of the last ice age look like a casual stroll through the park. We’re already warming the Earth about 20 times faster than during the ice age transition, and over the next century that rate could increase to 50 times faster or more. We’re in the process of destabilizing the global climate far more quickly than happens even in some of the most severe natural climate change events.
That rapid climate destabilization is what has climate scientists worried. It’s faster than many species can adapt to, and could therefore cause widespread extinctions, among other dangerous climate change consequences. Coastal flooding in places like Florida has already become much more common than it was just 50 years ago, and sea level rise is expected to keep accelerating.

Difference #2: Civilization. At the end of the ice ages we did not have 3 billion people living within 200 km from a coast line
That is not an difference in the cause of the warming, but definitely in the effect.
One could argue that that same civilization will make us able to deal with the consequences, but looking at thing that already exists several centuries without change, surely raises doubts to our ability to deal with the current rate of climate change. Think (1) unequal global wealth distribution or (2) the psychology of our behavior (denial, politics).

BTW The question asked to you

If climate change is not a natural phenomenon, then how did the last ice age end?

is based on a false premise - it's a bit of a word game.
The current warming is still a natural phenomenon, but it's now primarily driven by our input, and not other 'natural' phenomena.