Personally this is a no-brainer and the answer is AA. I moved to ORD a while ago and here are the reasons (in my view) AA is better if you live in the US:
1) You'll fly AA more often than you think. If flying to LIM why fly MSP-LAX-LIM or even with 2 stops if you can do MSP-DFW-LIM or MSP-MIA-LIM which will be shorter
2) You'll be gold already on AA. 25% mile bonus. MCE for free on t-24.
3) Booking awards is a breeze. And you can change the dates for free.
4) Lan's award chart is distance based if flying on AA. AA charges by zone, so award seats to LIM will be cheaper on average. And off peak saaver awards are super convenient compared to LAN awards. Even the more so if you don't live in MIA, JFK or LAX (LAN hubs) since LAN will charge you extra for the domestic US leg, AA won't.
5) No frustration in dealing with LAN. AA Gold phone agents know their stuff. Everything is predictable
6) Citi AA credit card
Downside:
1) Could AA move to a revenue based model soon? Maybe. Not in 2016 in my view.
2) LAN has tons of flights to LIM
3) Occasional promos and offers with KMs
4) LAN's award chart has some extra flexibility (i.e. awards are dynamically priced - AA doubles when going from sAAver to AAnytime)
Just my two cents.