Oh well that I did not know. I know many Catholics that believe they are different persons. But do you praise Mary then?
I don't know any Catholic who thinks that each of these are different persons. Every Catholic I've ever met recognizes that "Our Lady of _____" refers to Mary. Referring to her as "our lady" or "Madonna" is simply a title of respect. We bestow great honor on the woman who said "yes" to God, because of what she did, and because of who she was.
Now, if you believe that those who have died are not resurrected currently (e.g., that everyone who dies is just dead, and the resurrection won't happen until the future, at the Apocalypse, for everyone), then you wouldn't want to ask Mary, or any of the saints, to pray for you, because they're dead. Catholics, however, believe that these people are not asleep, but in heaven with God.
When we say, "Hail Mary...pray for us" we are not praying TO Mary, but we are asking her to pray for us, the same way you would ask a friend to pray for you if you needed prayers.
Mary is just a little closer to Jesus, because (a) she's already in heaven with him, and (b) as his mother, he's more likely to listen to her. Catholics do NOT believe that we have to pray to the Saints and Mary instead of God. Catholics are completely capable of praying to God directly. But hey, when you're in trouble, don't you want as many people praying for you as possible?
Hope that helps.