What is the difference between want to and care to ?Feel free to just provide example sentences.

I would say these mean the same thing. Usually 'care to' is used in a question i.e. "Do you care to have lunch?" But you could also say 'Do you want to have lunch?'
Many times 'care to' is used in future/potential questions phrased like "Would you care to have lunch?"

On the other hand, I could say "I want to have lunch" but it sounds unnatural to say "I care to have lunch"

In memrie's example above 'cares to' is replaceable with 'wants to' and the sentence would mean the same thing. If you wanted to draw attention to the 'caring' (emotional attachment), the sentence may read "He only cares about helping because..."

Just some examples. I just think most of the time they mean the same thing, and are sometimes interchangeable.