Both can be used, none is wrong. I think "never give on you" has a more informal ring to it, besides the fact that "give up" essentially means "desistir" and "give up on" has to do "desistir de uma causa, de um sonho, de algo maior, mais sério, ou mais duradouro."
It´s common the use of "give up on" as a "walk out on something/somebody" and "give up" as quit (something) or have done with/have something come to an end. Thus, "give up on" would mean to walk out and leave the other/the others move on with life (sometimes not so easy, but it´s an example here).
For example, someone could see the sentence "She gave the baby up because she wasn't ready to be a mother." (she gave up the idea of raising the baby -here an intersection with "give up on" but others could think otherwise, because to a certain degree she doesn´t have that lasting bonding with the baby yet. Anyway, if we are thinking in terms of short term (which is generally the case) then it´s "give up".
Other context is, someone that sees his relative/son etc won´t change and is a danger to society (a suspect criminal), so he "give up him/her" and turn him to the police/authorities.
So, depends on context.