“Empathize with” is when you have a deep understanding about how things actually feel for a person. You imagine yourself living life from the person’s perspective, as though you are inside their skin, experiencing everything through their eyes. (One idiom for empathy is “to be in someone’s shoes”. ) To relate to someone or something, is to understand (and agree with) but not on an emotionally-engaged level.

Empathize with is more associated with emotion while relate to is more associated with situations, Working weekends? I can relate to that. She's feeling lost. I empathize with her.