The worlds of graffiti and street art are exactly that; worlds apart.
Some graf artists become street artists, but street artists don't generally become graf artists.
From my dealings with both, I would say that both groups see themselves as achieving different things.
There are particular individuals from both groups who want nothing to do with the other. This further complicates the general public's view because of the general visual distinctiveness of each group: tags = graf, murals = street art.
If I were lucky enough to own property of my own, and someone wanted to visually express themselves on it, I would prefer them to come to me and ask. Maybe I would even collaborate. I suppose this statement would mean that I am not cool with the act of graffiti which happens anywhere and everywhere with no request. However I believe community spaces and the public domain is always open to negotiations between its participants, including those who feel the need to express themselves via graffiti.