There are some different opinions here, so here's mine: The words do have a similar meaning, because they both have to do with feeling something in the body. The sentences do not mean the same thing, as written. But if you write them this way, they have similar meanings: I clearly empathize with the people who live in those neighborhoods. The experience of the people who live in those neighborhoods clearly resonates with me. This story also attracted a lot of loving attention, and resonated with the public. This story also attracted a lot of loving attention, and the public empathized with the people in it. [The second sentence about the story makes sense grammatically, but sounds a little odd.] Empathy refers to feeling what another person is feeling. Something that resonates is anything that makes a person feel, it can be a situation, a thing, or a person.

you can empathise with someone, you can't resonate with someone. Something can resonate with you. So, 'the boy's story of losing his family resonated with me, as I had been in a similar situation. I empathised with him.'

resonate with -- I feel the same way as you do./I can relate to your experience. empathize with -- My opinion might be different from yours but I can see where you are coming from./I may not have experienced it but it must have been very difficult for you. I hope this helps.

I got it. Thank you Cathy.

They don’t mean the same thing. Resonate is more feeling similar to someone and empathy is more understanding someone’s difficulties. In your sentence pairs, I would say the first sentence is correct but not second. The third and not the fourth. Resonate in third sentence should be in past tense, resonated.