It depends on the material sometime
A vase is a ceramic, glass, pottery, "vessel"
An urn has a symbolic or religious purpose. It can also be a metal "vessel" with a lid for pouring liquids, such as tea or coffee.
It seems that "vessel" is the generic term for a hollow container regardless of it's use or the material from which it is made. That's what I thought, but looked it up to be sure. I took a pottery class or 2 back when, and made a few "vases".
The answer to your question will only be as detailed and specific as the question is detailed and specific. Good questions also include a sketch or a photo that illustrates your issue. (:< D)