In casual conversation, we usually support a claim with just a reason.

Example: We should leave (claim) because it looks like rain (reason).

We don't ask, "What evidence do you have that it looks like rain?" When you address serious issues in writing, though, you can't expect readers to accept all your reasons at face value. Careful readers ask for the evidence, the data, the facts on which you base those reasons.

Example: TV violence can have harmful psychological effects on children (claim 1) because those exposed to large amounts of it tend to adopt the values of what they see (reason 1 supporting claim 1 / claim 2 supported by reason 2). Their constant exposure to violent images makes them unable to distinguish fantasy from reality (reason 2 supporting reason 1 / claim 2). Smith (1997) found that children ages 5-9 who watched more than three hours of violent television a day were 25 percent more likely to say that most of what they saw on television was "really happening" (evidence supporting reason 2). 

At least in principle, evidence is something you and your readers can see, touch, taste, smell, or hear (or is accepted by everyone as just plain fact). It makes no sense to ask, "Where could I go to see your reasons?" It does make sense to ask, "Where could I go to see your evidence?"

• Reasons state why readers should accept a claim. Researchers can think up reasons; they don't think up evidence.
• Evidence is what readers accept as fact, at least for the moment. They think of evidence as "hard" reality, evident to anyone able to observe it.