In general usage, both have 2 common definitions, but only one of the uses overlap. When speaking about a lack of money, you can use either "poor" or "needy." However, "poor" is often used to show sympathy to someone who is hurting. If my little girl comes up with a scraped knee, I might say, "Oh, you poor thing!" I wouldn't use "needy" in this context. Another way to use "needy," however, is to talk about someone who always wants a lot of attention or relies too heavily on other people for comfort or other help. If I feel like I've recently asked for many favors from a friend, I might say, "I know I seem needy, but I just have one more thing that I need your help with."

I don't think there is much difference between these words. "poor" is the more common word, and "poor" can also mean "not up to standard" as well as "lacking money", whereas "needy" only means "in need of something".