Your question at first sounded obscene--"Sucking on Ice Cube (the rapper)"--until I read the rest of it. Be sure not to drop the final -s. It makes a difference. Both of your sentences basically mean the same thing, with the addition of the word "on" associated with the surface of something: "to suck on [the surface of] ice cubes," "to chew on [the surface of] ice." (By the way, chewing on ice is bad for your teeth.)

They both mean the exact same thing. I probably wouldn't say "Sucking ice cubes" because it sounds weird to me for some reason. Yet, they are both grammatically correct. When someone is "sucking on" something i envision their mouth cradled around the object, but if someone is "sucking" something, I envision them inhaling for some reason haha. With the exception of dicks. But we're not going to get into that lol.