Sometimes God calls us, or the circuмstances that God has placed us in, call us to perform charity work that no one else would see or appreciate, except God.  For example, it’s easy to give alms to the poor when they are nice people and when they are appreciative.  This type of charity has a built-in “reward” of seeing the good and of being appreciating by those whom you are helping.  

The type of charity that is truly heroic, and of which only catholic saints are able to do, through God’s grace, is that charity which has no earthly reward and has seemingly no “results”, or worse, it causes you pain/suffering.  Examples would be when heroic nuns would take care of the sick, and these patients would curse them and treat them horribly because they were angry at God for being sick and in pain.    

Or when St Therese the little flower suffered all manner of things under an older, jealous nun, yet St Therese treated her like her best friend, and eventually this nun converted because she saw the selflessness of St Therese.

Even Padre Pio suffered many, many things due to rumors and accusations of his character.  Many of the monks had mentioned a particular person who repeatedly called Padre Pio and fraud and a liar and stirred up trouble secretly, though Padre has so many friends that they eventually found out who was the trouble-maker.  One day this person showed up at the friary in the courtyard, as Padre Pio was with many of the monks, and Padre rushed over to him and embraced him (in the typical Italian way) saying he was happy to see his friend.  Padre did this to show his fellow monks that forgiveness must be immediate, complete and given over and over.  

When we do such works of charity, forgiveness and mercy, especially to those who hate/persecute us, this is the meaning of acting for “God’s sake”.  

What does it really mean when the Church says that we are to do anything for our neighbor "for God's sake"? Like give alms to people for God's sake. Does that mean that you shouldn't be thinking about the good you will do to someone when you help them? I read that that's the difference between Christian and non-Christian charity, that the latter only thinks about the good being done and not about it being done for God's sake.

Bingo.

Watch the movie Monsieur Vincent (1947), it is all about your subject. It is originally in French. I have it dubbed in English. Likely it also exists dubbed in Spanish. St. Vincent de Paul said that the poor will hate you for helping them.