There isn't any difference between "to wrap" and "to wrap up". The "up" can't even be really logically explained, which is perhaps why some people have not posted an answer. There is no concrete explanation or rule for when to add "up".
It's actually kind of nonsensical, because you're not applying the wrap in an upward direction, motion, or anything of the sort. It actually adds no additional information to the statement and does not change the meaning of anything. It's simply superfluous, unnecessary, and its addition is just a matter of personal preference.
The only additional implied meaning it can have in some situations is the implication of completeness. "Little Jimmy was all wrapped up in his blankets." By adding "up" here, it is loosely implied that Jimmy was entirely wrapped up in his blankets, covering him entirely. However, this does not concretely mean that if "up" was used here, then Jimmy must have been completely covered in his blankets.
Like I said, it's more a matter of personal preference to add "up" here, it doesn't hold any concrete meaning and is most likely technically incorrect, but widely used and acceptable to use.
To address the examples you provided. They are all technically correct/acceptable speech, but
Please wrap the tinfoil around the food.
would be perceived as very robotic. The reason why is because you're being very explicit with this sentence, instructing someone not just to wrap the food, but how to wrap the food with "around the food." It's very strict, explicit instruction that doesn't really need to be so explicit unless you are teaching someone how to wrap up food.