K2 presents much more difficult climbing at very high altitude. It has much worse weather, which is the greatest killer on these mountains. Ironically, the biggest hazard on Everest is at the lowest part of the climb, the Khumbu icefall. Everest on its easiest route has some spooky sections (the cornice ridge from the South Summit to the Hillary Step for example), but they are manageable with care. K2 climbing is a much higher level of risk. Here is the bottleneck at 8200 meters on the climb from Camp 4 to the summit, which requires the traverse of a 60 degrees snow/ice slope, which is at risk from overhanging seracs that do break and fall on the route.
Other portions of the route involve near vertical rock climbing still at high altitude. The Black Pyramid is between Camp 2 and 3, and looks like this:
The House Chimney is between Camp 1 and 2
That is just part of the chimney. All of this is more difficult and dangerous than anything on Everest, complicated by much more dangerous and unpredictable weather.
The mountains with the reputations for being the most dangerous are Annapurna and Nanga Parbat. 33% of all those who climbed Annapurna were killed on the mountain. Herzog and his party who were the first to summit nearly perished after being forced to bivouac at very high altitude. Very few have climbed the mountain, so the sample size is skewed. K2 death rate is 26%, and Everest about 4%, which is remarkable when taking into account the number of unqualified climbers on the mountain.