From Chile Wiki
The RUN (Rol Único Nacional) is a national ID number, unlike the RUT which is just a tax ID number. Some people get confused because, when you have a RUN, it will be used as your tax ID too. RUTs are for non-citizen non-residents, while the RUN is for Chilean citizens and foreign residents of Chile.
The confusion over the RUT / RUN difference is that at one time there was two different numbers for Chileans. Some time ago they where combined, and now the RUT number is used for almost every form of official document including national ID, passport number for Chilean citizens, even tracking of customer orders at private businesses.
So, your national ID number can be interchangeably called a RUT or a RUN by Chileans, but is most typically just called a RUT number. Really, now the only difference is how the number was issued to you in terms of your day to day activity in Chile. For a native Chilean they receive it at birth. For a foreigner asking for residency in Chile, you receive the RUT from the IRS that later is converted to a RUN for your national ID.