If you fly into Chile on a US passport you owe $131 (currently) as a reciprocity fee to the Chilean government. To fly into Argentina you owe $0 and Buenos Aires is a much, much, much nicer city. Cheap wine, women and beef. From either city you can get flights to the south or, if you've a couple days extra time and less money, comfy long distance buses(think barcalounger with meals and movies!). Booking in advance depends what time of year you are going - holidays can be a pain to get things last minute, but the price advantage isn't much.
Punta Arenas = easy access to T del Paine. If you want Fitzroy region easier to fly into Rio Gallegos (3 or so hours from Buenos Aires) and take a busy to El Calafate, then a bus to El Chalten (should be better with the paved roads)
Airmiles? What airline - I've flown to Buenos Aires 2x on miles (northwest and alaska miles, not the metal of either) and in August Santiago (alaska miles again).