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."