Sometimes they can be stated in a way that can be the very same thing. Many times it will be different, very often indeed.
Thus, I am going to explore the second option.
Conflicting - may have to do with opposing and different in kind. One thing that can´t (almost) can´t
cohexist with another, if so it can´t cohexist in a pacific way.
Contradictory - may have to do with something ''inconsistent/logically opposite", it´s said about a proposition(s), or act(s) that are a contradiction in nature (e.g. do or say something -seemingly- in denial to something said by you some time before...), revealing inconsistency of action or ideas etc.
It´s easier to understand that other people may have conflicting interests (in a way, the person is inherently suspect of being biased towards one or the other interest): one example, the tobaco industry, wich have to sell cigarretes and then having to show the effects of its product, they put the photos of
people with their health harmed by smoking cigarretes. There is a conflict there, but the industry can do with that.
An example of contradictory situation, a woman says that hates some friend of her, that before her child. Then the friend comes and knock the door, the woman peck her cheek, ask her to come in and drink tea with cookies...it´s contradictory to the child...to him/her it doesn´t makes sense at all !