Can/can’t indicate ability while will/won’t indicate willingness.
What’s the difference between ability and willingness? Ability means you have the skills. Willingness means you want to.
Look at these examples and see if you can get a feel for the difference:
This woman can help you. She has the information that you need. Unfortunately, she’s in a bad mood, so she won’t. She doesn’t feel like being helpful. She will make you wait until you give up and leave her alone so she can go back to drinking coffee and watching YouTube.
These students can answer the teacher’s questions. They have the information right in front of them. However, they are bored. They hate school, so they won’t. They‘ll look at their cell phones until their teacher can’t stand it any more.
This woman swears that she can drive. She says she hasn’t had too much to drink. However, her friends won’t let her drive. They believe she can’t drive safely. They won’t let her endanger herself and others. Instead, they will call her a cab. They‘ll even pay for it.