When we are talking about people (as opposed to things), these words are essentially synonyms. But there is, perhaps, some nuance: If something "resonates" with a person, it has an emotional connection with them. In some way, it "sings" in them. Notice that the -son- part of "resonate" means "sound" (from the Latin "sonus"—compare "sonic"). It is a deeper word. If we "relate" to something, it means that we feel that it is similar to our experience. This is like "resonate," except more focused on experience. Consider this example of both words: I love that movie. It was so profound— the cinematography, the writing, the acting, the music. It all really resonated with me. The main character was so human! I felt like I could relate to her, since I have had similar experiences in my life. Now, "resonate" is not usually used with things (for a thing to resonate with a thing). What I mean is that "this movie resonates with that movie" is not proper English. But with "relate," we can speak of a connection between things — the relationship between them. For example, a movie can relate to another movie in many ways — maybe they're in the same genre, or have the same director, or are in a series. Your question was clear. I might add "quotes" to make it even clearer: What’s the difference between "resonate with someone or something" and "relate with someone or something"? I hope that helps!

Have you ever tuned a guitar, or similar instrument, using a tuning fork? If, say, your G string is in tune then, when you strike a G tuning fork the G string of your guitar will start to vibrate all by itself even though you have not touched it. It starts to "resonate". Look at the roots of that word. "Resonate" means "to sound again". It is the same with people. "Resonate" always creates a musical image. If Sally resonates with Becky it means that Becky's "vibes" rub off on Sally. Sally comes to life around her because they are "in tune" with each other. "Relate" has no such poetry. It is not nearly as beautiful a word. It creates no imagery.

Your question: "What's the difference between saying 'I resonate with that' and 'I can relate to that'?" - The way you wrote it is fine for written English. For spoken English, the way I wrote it is a little more natural, at least for me. As for the actual difference, Eric gave a great explanation that I can't top, and won't bother trying. :)