This will be my eighth season in Chile and Argentina. I am there for anywhere from 10 weeks to four months every year. In many ways it is like a home to me, and I have skied in many places from Portillo to Bariloche. I won't speculate which is better. Both South America and NZ are stunning, and the experience will blow you away - if you want it to. You can't lose with these choices.
That said, I may be able to share some information about Chile and Argentina.
Are you looking for a purely resort experience? Only backcountry? A little of each? The areas near Santiago - Valle Nevado and its neighbors being one-hour away, and Portillo being two-ish-hours away - are well established resorts with decent infrastructure. These places are situated high in the mountains, above treeline, and are pretty remote. The primary activity will be skiing, as there are no real towns or anything to really keep you entertained on a "day off." Both of these locations offer a mix of resort runs and access to the backcountry. In both areas, heli-skiing is available for a day, or even just a morning. Portillo is smaller, but all of the terrain is fun. There is only one hotel in Portillo, so you pay their price and live with the culture there. It is an old, classic hotel, with an established rhythm. Most people love it, some people do not.
Nevados Chillan is further south and also has a mix of resort and backcountry. A lot of people like it because there is also the town of Las Trancas, which adds to it's South American feeling. The resort has some serious ownership issues, and some seasons the best lifts don't run because of territorial pissing. It is a bit lower, and offers some tree skiing. It can also rain at any time of year.
Las Leñas is a big mountain several hours south of Mendoza, Argentina. There is little debate that this is the best lift-served terrain in South America. But you can pay the price for a chance to win it all. It is more challenging to get there - there are multiple options, and none of them are easy. It is more likely that you will encounter high wind than deep snow. The upper lift, the Marte, can stay closed for days. There are more hotel options in the valley than in a place like Portillo, but it's still not really cheap. But there's a little more going on, and as a young man just out of the military academy, the girls visiting from Mendoza will make you never want to return home - I am not sure what is in the water there. It's not quite all or nothing there, but it's close. I think most adventurous people feel Las Leñas is worth the gamble.
Bariloche is in the northern part of Argentine Patagonia. It is lower, but most of the skiing is above treeline. There is a well established resort, as well as a fun town. There is a ton going on all winter. The skiing can be excellent, but because it is low, it can also really suck. There is a hut system behind the resort, so there is easy access to some fun touring. I know a bunch of gringos that love visiting Bariloche when they only have a week. It appears logical to get there by going through Buenos Aires, but I think it's easier from Santiago. If you can't fly to Bariloche due to budget, you can bus it fairly simply from either capital city.
Where you go can depend on when you go. As a general rule, the earlier in the season you go, the further north you want to be. So in July, the areas around Santiago are your best bet. In August, any of the above mentioned places are fairly consistent - or they all have a chance of being great. In September, I think Bariloche can be a little too warm, although I've seen photos of people hitting right. Las Leñas is typically spring-like, which is generally good for them. In September, Chilean Patagonia becomes an option if you want to ski tour. All of the volcanoes have small resorts on their flanks, but these really aren't worth anything if you're from the northern hemisphere. The ski touring on these volcanoes is really fun though. You can link these up into amazing, well-rounded vacations.
Chile is a little more expensive than Argentina. The service is also a little more consistent. By this I mean things like avalanche control work and chairlifts work a little more "normally." The bus systems of both countries are fantastic. It is easy to get around - Chile especially. Both nations are full of friendly, mostly helpful people. English is not always spoken, so some basic spanish, or a lot of patience, is helpful. I feel I get less run around in Chile, but I have spent much more time there. Argentine Spanish is a different accent, but it is spoken clearly. Chilean Spanish is the equivalent of English in Louisiana - it can be challenging to understand.
If you'd like to ask more questions or share more of your details, I will be happy to help. I am a guide, and as I said I spend most of each southern winter there. But I won't sell you anything. I truly love the country and the mountains and love to help people get down for whatever adventure they're after.