In my opinion, these differ somewhat. The phrasal verb "check up" has the meaning to investigate. It has the connotation that someone might have done something wrong. A suspicious wife or girlfriend might call to "check up" on a man who could be engaged in outside activities. On the other hand, if I call to check on someone, it is probably a friendly call to verify well-being. I might say, "Hello, Mother! I just called to check on you." I would not say, "Mother, I am calling to check up on you." My dear mother would never do anything wrong.

There isn't much of a difference. In both instances, you're calling to see how somebody's doing in your absence.

The only difference would be that using the word "up" comes across more friendly, as a family member or close friend might say. Without it, it's still very friendly, but it may be how a boss or school teacher calls to see how you are doing.

In your first sentence "up" is just metaphorical for saying "on". It's just a slang way of saying the same thing. Good luck : )

I have no opinion, I am a new learner,