I know that "something that interests me" and "something interesting me" are equivalent. But I would like to know if there is any further subtle difference between these two?
They're related, but they're not the same.
"Something that interests me" is talking about the thing itself. "What are you staring at?" "I'm looking at that building. Architecture is something that interests me."
"Something interesting me" refers to the process, not the thing. I'm struggling to come up with an example that does not sound awkward because it's not something a fluent English speaker would normally say. Maybe, "Why did you walk into the fence?" "I was distracted, my mind was occupied by something interesting me." But that's awkward.
Something that interest me vs Something interesting me.
They are not at all similar. "Interest" has been used as a verb in the first sentence, whereas "interesting" is an adjective in the second one. They will convey the same sense if used in the following way:
Something that interests me / Something that is interesting to me.