The biggest difference between the Andes and the Alps is the dense collection of all different kind of rocktype (f. e. limestone in the Dolomites, granit and andesit in Mont Blanc range) within one big mountainrange. Totaly different looks are the result.
The Mont Blanc range is highly packed with glaciers because of it`s elevation, geographical location and it's 4 kilometer vertical dropdown out of flat lands to the west of the range. The Dolomites have almost no glaciers, because they are lower. Same with German Alps. This makes the Alps special for any visitor, because there is so much variety to see in rather short distance.
The valleys in the Alps are not comparable to anything I`ve seen in Southamerica, because the Alps have a totally different flora.
There is rainforest in Venezuela, there are almost no trees in Peru, Equador and the northern parts of Chile and Argentina, there are great beech-woods in Patagonia.
But I haven't seen any mountain in the Andes, that is packed with glaciers and ice for a kilometer of vertical distance, followed by another vertical kilometer of rocky belt and finally a third vertical kilometer of conifer forest all the way down into a valley covered with green grass.
White (snow), green (forest) and ginger (rocks) are the dominating colours in a great balance.
Another difference are the glaciers for sure. The Andes have rugged mountains in Peru (f. e. Cord. Blanca) but those peaks are not highly surrounded with longer glaciers, because of their climatic location. The glaciers are reduced to areas upper 5000 meters.
To find comparable glaciers you have to go to Patagonia. The glaciers there are even bigger than the ones in the Alps, but the mountain ranges are only few - and from size in square kilometers rather small and dotted local groups - like Torres del Paine, Fitz Roy range or Cerro San Lorenzo.
Don`t get me wrong: the Andes - especially in Patagonia - are fascinating! But they are different from the Alps in many, many ways more than what I shortly described here.
I've seen some pictures and movies from NZ. If anything can be compared with the character of the European Alps (rocktype, glaciers, flora, elevation), then this is the closed match.